Module 3 Version 03 Sections 1 3. Math 7. Module 3. Lines and Shapes. 5 cm. 6 cm. 10 cm. 100 b

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1 Module 3 Version 03 Sections 1 3 Math 7 Module 3 Lines and Shapes 6 cm 5 cm 10 cm 6 cm 10 cm h 100 b

2 2008 by Open School BC This work is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License. To view a copy of this license, visit Course History New, September 2008 Project Manager: Jennifer Riddel Project Coordinator: Eleanor Liddy, Jennifer Riddel Planning Team: Renee Gallant (South Island Distance Education School), Eleanor Liddy (Open School BC), Steve Lott (Vancouver Learning Network), Jennifer Riddel (Open School BC), Mike Sherman, Alan Taylor (Raven Research), Angela Voll (School District 79), Anne Williams (Fraser Valley Distance Education School) Writers: Meghan Canil (Little Flower Academy), Shelley Moore (School District 38), Laurie Petrucci (School District 60), Angela Voll (School District 79) Reviewers: Daniel Laidlaw, Steve Lott (Vancouver Learning Network), Angela Voll (School District 79) Editor: Shannon Mitchell, Leanne Baugh-Peterson Production Technician: Beverly Carstensen, Caitlin Flanders, Sean Owen Media Coordinator: Christine Ramkeesoon Graphics: Cal Jones

3 Module 3 Table of Contents Course Overview 3 Module Overview 8 Section 1: Geometric Constructions 9 Pretest 11 Lesson A: From Points to Lines 17 Lesson B: Tools and Terminology 39 Lesson C: Constructing Perpendicular Bisectors 49 Lesson D: Constructing Angle Bisectors 65 Lesson E: Constructing Parallel Line Segments 77 Section 2: Areas of Polygons 93 Pretest 95 Lesson A: How Do We Measure Things? 99 Lesson B: Finding the Area of Triangles 117 Lesson C: Finding the Area of Parallelograms 131 Lesson D: Problem Solving with Area 145 Section 3: Circles 165 Pretest 167 Lesson A: What s in a Circle? 173 Lesson B: What is Pi? 187 Lesson C: Circumference of a Circle 197 Lesson D: Area of a Circle 211 Templates 229 Answer Key 239 Glossary 265 Open School BC MATH 7 etext Module 3 1

4 2 MATH 7 etext Open School BC Module 3

5 Course Overview Welcome to Mathematics 7! In this course you will continue your exploration of mathematics. You ll have a chance to practise and review the math skills you already have as you learn new concepts and skills. This course will focus on math in the world around you and help you to increase your ability to think mathematically. Organization of the Course The Mathematics 7 course is made up of seven modules. These modules are: Module 1: Numbers and Operations Module 2: Fractions, Decimals, and Percents Module 3: Lines and Shapes Module 4: Cartesian Plane Module 5: Patterns Module 6: Equations Module 7: Statistics and Probability Organization of the Modules Each module has either two or three sections. The sections have the following features: Pretest This is for students who feel they already know the concepts in the section. It is divided by lesson, so you can get an idea of where you need to focus your attention within the section. Section Challenge This is a real-world application of the concepts and skills to be learned in the section. You may want to try the problem at the beginning of the section if you re feeling confident. If you re not sure how to solve the problem right away, don t worry you ll learn all the skills you need as you complete the lessons. We ll return to the problem at the end of the section. Open School BC MATH 7 etext Module 3 3

6 Each section is divided into lessons. Each lesson is made up of the following parts: Student Inquiry Inquiry questions are based on the concepts in each lesson. This activity will help you organize information and reflect on your learning. Warm-up This is a brief drill or review to get ready for the lesson. Explore This is the main teaching part of the lesson. Here you will explore new concepts and learn new skills. Practice These are activities for you to complete to solidify your new skills. Mark these activities using the answer key at the end of the module. At the end of each module you will find: Resources Templates to pull out, cut, colour, or fold in order to complete specific activities. You will be directed to these as needed. Glossary This is a list of key terms and their definitions for the module. Answer Key This contains all of the solutions to the Pretests, Warm-ups and Practice activities. 4 MATH 7 etext Open School BC Module 3

7 Thinking Space The column on the right hand side of the lesson pages is called the Thinking Space. Use this space to interact with the text using the strategies that are outlined in Module 1. Special icons in the Thinking Space will cue you to use specific strategies (see the table below). Remember, you don t have to wait for the cues you can use this space whenever you want!? Just Think It: Questions Write down questions you have or things you want to come back to. Just Think It: Comments Write down general comments about patterns or things you notice. Just Think It: Responses Record your thoughts and ideas or respond to a question in the text. Sketch It Out Draw a picture to help you understand the concept or problem.! Word Attack Identify important words or words that you don t understand. Making Connections Connect what you are learning to things you already know. Open School BC MATH 7 etext Module 3 5

8 More About the Pretest There is a pretest at the beginning of each section. This pretest has questions for each lesson in the sections. Complete this pretest if you think that you already have a strong grasp of the topics and concepts covered in the section. Mark your answers using the key found at the end of the module. If you get all the answers correct (100%), you may decide that you can omit the lesson activities. If you get all the answers correct for one or more lessons, but not for the whole pretest, you can decide whether you can omit the activities for those lessons. Materials and Resources There is no textbook required for this course. All of the necessary materials and exercises are found in the modules. In some cases, you will be referred to templates to pull out, cut, colour, or fold. These templates will always be found near the end of the module, just in front of the answer key. You will need a calculator for some of the activities and a geometry set for Module 3 and Module 7. If you have Internet access, you might want to do some exploring online. The Math 7 Course Website will be a good starting point. Go to courses/math/math7/mod3.htm and find the lesson that you re working on. You ll find relevant links to websites with games, activities, and extra practice. Note: access to the course website is not required to complete the course. 6 MATH 7 etext Open School BC Module 3

9 Icons In addition to the thinking space icons, you will see a few icons used on the lefthand side of the page. These icons are used to signal a change in activity or to bring your attention to important instructions. Explore Online Warm-up Explore Practice Answer Key Use a Calculator Open School BC MATH 7 etext Module 3 7

10 Module 3 Overview As its title suggests, this module is all about lines and shapes. You ll need to get out your pencil, ruler, protractor, and compass, because this module is truly hands-on! You ll draw lines, angles, and all kinds of shapes. You may remember, from previous grades, how to calculate the perimeters and areas of some shapes. In this module, you ll review those topics and then learn the skills you need to find the areas of other shapes, like triangles and parallelograms. Once you ve explored polygons, it will be time to move to circles. You ll investigate many of the characteristics of circles and learn about a special number called pi (π) are you hungry yet? Section Overviews Section 3.1: Geometric Constructions In this first section of Module 3, you will gain a working understanding of lines and angles. You will learn some new terminology, and then it will be time to get out the toolbox! Armed with a protractor and a compass, you ll be performing a number of geometric constructions, all on your own. By the end of this section you ll have all the skills you need to design a tree house! Section 3.2: Areas of Polygons In this section, you ll build on your knowledge of measurement. You ll discover how to find the area of a triangle and of a parallelogram. Then, applying these new skills, you ll tackle some everyday problems. Put your thinking cap on, these problems might have a few steps but don t worry, you ll be well prepared to solve them! Section 3.3: Circles By the time you move on to this third section, you ll be pretty familiar with lines, angles, polygons, and with measuring all of these things. There s one shape left in this module: the circle. In this section you ll investigate the characteristics of circles and discover a special relationship between the parts of a circle. This relationship will help you measure the area of, and the distance around, any circle you meet. 8 MATH 7 etext Open School BC Module 3

11 Section 3.1: Geometric Constructions Contents at a Glance Pretest 11 Section Challenge 16 Lesson A: From Points to Lines 17 Lesson B: Tools and Terminology 39 Lesson C: Constructing Perpendicular Bisectors 49 Lesson D: Constructing Angle Bisectors 65 Lesson E: Constructing Parallel Line Segments 77 Section Summary 87 Learning Outcomes By the end of this section you will be better able to: find examples of parallel line segments and perpendicular line segments in the world around you. draw parallel line segments and perpendicular line segments and explain why they are parallel or perpendicular. draw a perpendicular bisector and explain what makes it a perpendicular bisector. draw an angle bisector and explain how you know it s an angle bisector. Open School BC MATH 7 etext Module 3, Section 1 9

12 10 MATH 7 etext Open School BC Module 3, Section 1

13 Pretest 3.1 Pretest 3.1 Complete this pretest if you think that you already have a strong grasp of the topics and concepts covered in this section. Mark your answers using the key found at the end of the module. If you get all the answers correct (100%), you may decide that you can omit the lesson activities. 1. Sort the following groups of lines into parallel or perpendicular lines. a. b. c. d. Open School BC MATH 7 etext Module 3, Section 1 11

14 2. Draw a line segment of the length given. Then, use your protractor to draw the angle on the line segment. a. Line segment = 4 cm, angle = 45 b. Line segment = 3 cm, angle = 120 c. Line segment = 2.5 cm, angle = List three examples of perpendicular lines in everyday life List three examples of parallel lines in everyday life MATH 7 etext Open School BC Module 3, Section 1

15 5. Fill in the blanks. Pretest 3.1 a. You can use a to construct circles and arcs. b. A intersects a line segment at 90 degrees and divides it into two equal lengths. c. One example of a perpendicular bisector is the letter. d. You can use a to measure angles. e. An cuts an angle in half to form two equal angles. 6. Draw a perpendicular bisector of a line segment using different construction methods. For each question, draw a line segment that is the length indicated. Then construct the perpendicular bisector using the given construction method. a. 73 mm compass and straight edge (ruler) Open School BC MATH 7 etext Module 3, Section 1 13

16 b. 14 cm protractor and ruler c. 11 cm right angle and ruler 14 MATH 7 etext Open School BC Module 3, Section 1

17 7. Use a protractor to construct each angle. Then construct the angle bisector. Use a different method of construction for each angle. Pretest 3.1 a. 130 angle b. 70 angle 8. Draw two parallel line segments that are each 5 cm long using the compass and ruler method. 9. Draw a rectangle with length (6 cm) and width (3 cm) using the protractor and ruler method. Turn to the Answer Key at the end of the Module and mark your answers. Open School BC MATH 7 etext Module 3, Section 1 15

18 Section Challenge Sarah and Tom want you to design a tree house for their backyard. It sounds very exciting and you have lots of ideas. Quickly, you realize that there are many things that you will need to consider. To start, you ll have to choose a suitable tree. You ll also need to know how to draw parallel and perpendicular line segments. The floor of the tree house has to run parallel to the ground so that people can stand up straight. Also, the walls in the tree house have to run perpendicular to the floor, so they won t fall over. You ll need to design and build a ladder so they can climb up to their tree house. Sarah and Tom s friends also want you to draw a map with an arrow pointing out the direction to the tree house, so they know how to find it. If you re not sure how to solve the problem now, don t worry. You ll learn all the skills you need to solve the problem in this section. Give it a try now, or wait until the end of the section it s up to you! 16 MATH 7 etext Open School BC Module 3, Section 1

19 Lesson 3.1A: From Points to Lines Lesson 3.1A Student Inquiry What are some real life examples of lines? This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 1 17

20 Student Inquiries BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: What are parallel lines? answer example Can I find an example of parallel line segments in the world around me? answer example What are perpendicular lines? answer example Can I find an example of perpendicular line segments in the world around me? answer example 18 MATH 7 etext Open School BC Module 3, Section 1

21 Lesson 3.1A: From Points to Lines Introduction A long time ago in Ancient Egypt, a mathematician named Euclid wanted to understand the world around him and prove that things were true by using logic and reason. He did this by studying geometry and the properties of angles, points and straight lines, and shapes such as triangles, squares, and circles. Lesson 3.1A Thinking Space Euclid wrote books on geometry that have been studied for over 2000 years by math students, artists and architects. Like the Ancient Egyptians, we will begin studying geometry starting with Euclid s observation: You can draw a straight line between any two points. Let s begin! Note: You will need a geometry set to complete the activities in this lesson. Explore Online Looking for more practice or just want to play some fun games? If you have internet access, go to the Math 7 website at: Look for Lesson 3.1A: From Points to Lines and check out some of the links! Open School BC MATH 7 etext Module 3, Section 1 19

22 Warm-up Let s first review some of the terminology and rules you already know about geometry. Try to fill in the blanks below using the following words: parallel meet perpendicular degree right protractor acute obtuse 3 o clock zero 1. An angle is formed when two lines. 2. The unit for measuring angles is the. 3. lines never meet. 4. A protractor has 180 congruent slices. Each slice is 1 degree. You write At the hands of the clock make an angle of 90 degrees. 6. lines meet at A 90 angle is called a angle. 20 MATH 7 etext Open School BC Module 3, Section 1

23 8. The measure of an angle is less than 90. Lesson 3.1A 9. The measure of an angle is between 90 and You use a to measure angles. 11. You always read your protector starting from. 12. Just by looking at these angles, see if you can tell which ones are 90, which ones are less than 90, and which ones are greater than 90. a. b. c. d. e. f. Turn to the Answer Key at the end of the Module and mark your answers. Open School BC MATH 7 etext Module 3, Section 1 21

24 Explore How to Use a Protractor Thinking Space Here is an illustration of a protractor, a tool that you will find in your geometry set. scale measures from the left scale measures from the right starts at 0 centre mark base line starts at 0 You will see that there are two scales on each protractor. One scale starts at the right. The other scale starts at the left. Both scales are marked from 0 through 180. Now let s review how to use the protractor to measure angles. Step 1: Step 2: Place the centre mark of the protractor directly on the vertex of the angle. Line up one arm of the angle with the base line of the protractor so that it starts at 0 degrees. Step 3: Use the scale that starts at 0. Step 4: Read the measure of the angle. 22 MATH 7 etext Open School BC Module 3, Section 1

25 Example 1: Look at the protractor shown below. Using the steps on how to use a protractor, you should find that the angle is 40. Lesson 3.1A Thinking Space Step 4: Read the measure here. Step 1: Vertex Step 3: Use this scale. Step 2: Line up line segment with base line Example 2: Look at the protractor shown below. Using the steps on how to use a protractor, you should find that the angle is 50. Open School BC MATH 7 etext Module 3, Section 1 23

26 Example 3: Look at the protractor shown below. Using the steps on how to use a protractor, you should find that the angle is 140. Thinking Space Now it s your turn. 24 MATH 7 etext Open School BC Module 3, Section 1

27 Practice 1 Lesson 3.1A 1. Read these protractors to give the size of each angle shown. a. b. Angle = Angle = c. d. Angle = Angle = Open School BC MATH 7 etext Module 3, Section 1 25

28 2. Use your protractor to measure these angles A B C D E A = B = C = D = E = Turn to the Answer Key at the end of the Module and mark your answers. 26 MATH 7 etext Open School BC Module 3, Section 1

29 Explore Now you are going to learn how to draw your own angles. Remember Euclid s observation: you can draw a straight line between any two points. Lesson 3.1A Thinking Space Line Segments, Lines, and Rays A line segment is a straight line between two points. It has two definite end points and a definite length. AB means the line segment from A to B. A B A line is a straight mark. A line is continuous without a beginning or end. A B A ray has a beginning point, but no end. A What are the differences and similarities among line segments, lines and rays? Open School BC MATH 7 etext Module 3, Section 1 27

30 How to Draw an Angle Using a Ruler and Protractor Here are the steps to follow: Thinking Space Step 1: Start with two points. A B Step 2: Using your ruler, draw a line between these two points. A B Step 3: Place the centre of the protractor on one point of the line segment A 0 B Step 4: Line up the line segment with the base line of the protractor. Step 5: Start at 0 on the base line. Step 6: Count around from 0 until you reach the angle you would like to draw. Step 7: Make a mark with your pencil on the paper A 0 B Step 8: Remove the protractor. Step 9: Draw a line from the point at the centre of the protractor to the mark on your paper. A B Step 10: Label the measured angle. C 30 A B 28 MATH 7 etext Open School BC Module 3, Section 1

31 Draw an Angle with your Protractor Now it is your turn to draw an angle: Starting with vertex L on LM draw a LN with an angle of 60 from LM. Lesson 3.1A Thinking Space L M Starting with a vertex T on ST draw an angle of 135. S T Open School BC MATH 7 etext Module 3, Section 1 29

32 Starting with a vertex X on XY draw an angle of 90. Thinking Space X Y Check to see that your angles match the ones here. N L M S T X Y 30 MATH 7 etext Open School BC Module 3, Section 1

33 Here s some new terminology relating to intersecting lines. Classification of Angles These are some terms that we use to classify angles by size. You might know some of them already. Lesson 3.1A Thinking Space Right Angle Measures exactly 90 (shown by a small box in the corner.). Acute Angle Measures less than 90. Obtuse Angle Measures more than 90 but less than 180. Straight Angle Measures exactly 180. Reflex Angle Measures more than 180 but less than 360. Open School BC MATH 7 etext Module 3, Section 1 31

34 Parallel and Perpendicular Lines These two line segments are parallel. Thinking Space The sides of a ruler are parallel so are the sides of a ladder. Hopefully, the sides of your house are parallel to each other! What does parallel mean? Parallel lines can be extended forever in both directions and they will never cross.! These lines are parallel. These lines are not parallel. When these lines are extended, they intersect The symbol,, means parallel. AB CD means line AB is parallel to line CD. 32 MATH 7 etext Open School BC Module 3, Section 1

35 These two lines are perpendicular. Lesson 3.1A Thinking Space The top of a picture frame is perpendicular to the side. The rungs of a ladder are perpendicular to the sides of the ladder. In your house, the walls are perpendicular to the floor well, they should be! What does perpendicular mean? Perpendicular lines meet at right angles. Check the examples above with your protractor. When you find an angle that is 90, you can mark it like this:! A right angle is a 90 angle. The symbol shows us that the angle is a right angle; it measures exactly 90. Open School BC MATH 7 etext Module 3, Section 1 33

36 Use your protractor to measure the angles below. Mark the right angles using the symbol. Thinking Space You should have marked the second and fourth angles like this: We have another symbol that we use when two lines are perpendicular. It looks like this: AB CD means line AB is perpendicular to line CD. 34 MATH 7 etext Open School BC Module 3, Section 1

37 Here s another one of Euclid s observations: All right angles are equal. If two lines intersect in such a way that they form four congruent angles, they are said to be perpendicular to each other. The angles formed are called right angles (90 ). Lesson 3.1A Thinking Space C A B D The symbol for right angle is SYMBOLS The symbol for perpendicular to is Open School BC MATH 7 etext Module 3, Section 1 35

38 Practice 2 Many things in everyday life remind us of parallel lines and perpendicular lines. The rails of a railroad or the sides of a ruler remind us of parallel lines. The corners of a picture frame or the corners of a room remind us of perpendicular lines. 1. Sort the following groups of lines into parallel or perpendicular lines. a. b. c. d. Parallel: Perpendicular: 36 MATH 7 etext Open School BC Module 3, Section 1

39 2. Use your protractor and check which angles are right angles (90 ). Lesson 3.1A a. b. c. Right angles: 3. Use your protractor to draw the following angles on the given line segment. a. b. 30º 45º c. d. 60º 90º e. 180º Open School BC MATH 7 etext Module 3, Section 1 37

40 4. List three examples of perpendicular lines in everyday life: List three examples of parallel lines in everyday life: Turn to the Answer Key at the end of the Module and mark your answers. 38 MATH 7 etext Open School BC Module 3, Section 1

41 Lesson 3.1B: Tools and Terminology Lesson 3.1B Student Inquiry Looks like I ll need my geometry set! This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 1 39

42 Student Inquiries What are some examples of perpendicular bisectors in the world around me? What are some examples of angle bisectors in the world around me? BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: answer example answer example 40 MATH 7 etext Open School BC Module 3, Section 1

43 Lesson 3.1B: Tools and Terminology Introduction Have you ever wondered how bridges, buildings, railroad tracks, picture frames, soccer fields, or quilts are designed and constructed? Builders all start with similar tools that you can find in your geometry set. In this lesson, we will look at the different tools used in geometry. We will also learn some new words related to geometry. Once you know the definitions, you ll be able to use these words to easily explain the many different lines, angles, and shapes you draw in geometry. Lesson 3.1B Thinking Space Open School BC MATH 7 etext Module 3, Section 1 41

44 Warm-up Geometry is the study of points, lines, angles, and shapes. What kinds of tools can you use to construct points, lines, angles, and shapes? What are some tools that carpenters and engineers use to construct lines and angles in buildings with? Using a web, brainstorm some ideas. Start by writing the words geometry tools in the middle of the circle. Think of as many tools as you can. Here s one to start with: What tool do you need to draw a straight line? Geometry Tools ruler (straight edge) 42 MATH 7 etext Open School BC Module 3, Section 1

45 Explore The Ancient Egyptians were great builders they constructed pyramids, sculptures, temples, and tombs. To build the pyramids they used a straight edge and a rope compass. Land surveyors, architects, engineers, and sculptors in Ancient Egypt all used the rules of geometry in their trade as people do today. Using a compass and straightedge is still the best way to do construction. Think of what our houses would look like if carpenters didn t have tools to make straight edges, vertical walls, and square corners. The walls would not stand up straight, and the house might fall over, like a house built out of a deck of cards! Lesson 3.1B Thinking Space? Do you have any questions about the Ancient Egyptians or the techniques they used? Here are some important geometry tools and terminology: Protractor You can use it to measure angles. Compass You can use it to construct circles and arcs. Straight edge and/ or ruler You can use it to draw a straight line. 45 right angle You can use it to draw 45 angles and right angles (90 ) right angle You can use it to draw 30, 60, and 90 angles. Square or T-square You can use it to draw right angles Open School BC MATH 7 etext Module 3, Section 1 43

46 Intersection point The point where two lines cross each other. Intersection point Thinking Space Perpendicular bisector A line that intersects a line segment at 90 and divides it into two equal lengths. 90 } }1.5 cm 1.5 cm Perpendicular bisector Angle bisector A line that cuts an angle in half to form two equal angles Angle bisector Can you think of any examples of these terms in your daily life? Here are some examples: Framing Square and Level Used to check that the wall is perpendicular to the floor. Intersection Point Pedestrians must stop at the street intersection before crossing. Perpendicular Bisector The letter T is a line segment with a perpendicular bisector. Angle Bisector The fold line when you fold an equilateral triangle in half. 44 MATH 7 etext Open School BC Module 3, Section 1

47 Practice Lesson 3.1B Let s practise constructing shapes with your geometry tools. 1. Practise using the different geometry tools. Discover the many types of shapes you can make on your paper. a. Can you draw a flower with petals using your compass? b. Can you draw a 30, 45, 60 angle using only the triangles? Try measuring the angles with your protractor to see if you are correct. c. Can you draw a star? Open School BC MATH 7 etext Module 3, Section 1 45

48 2. Ancient Egyptians built the pyramids using a rope compass and straightedge. Architects commonly use a square or triangle to draw a right angle. Can you think of some other situations where geometry tools would be useful? Try and write down at least three ways to use geometry tools Can you think of examples of perpendicular bisectors in everyday life (eg. A teeter-totter in the playground)? Can you describe examples of angle bisectors in the environment (eg. A pole holding up a triangular shaped tent)? Let s review the new terminology that you have learned. 4. Fill in the blanks. Hint: if you re stuck, go back to the vocabulary table from earlier in the lesson. a. You can use a to construct circles and arcs. b. A intersects a line segment at 90 and divides it into two equal lengths. 46 MATH 7 etext Open School BC Module 3, Section 1

49 c. One example of a perpendicular bisector is the letter. Lesson 3.1B d. You can use a to measure angles. e. You can use a to measure 30, 60, or 90 angles. f. You can use a to draw a straight line. g. The is where two lines cross each other. h. An cuts an angle in half to form two equal angles. Turn to the Answer Key at the end of the Module and mark your answers. Open School BC MATH 7 etext Module 3, Section 1 47

50 48 MATH 7 etext Open School BC Module 3, Section 1

51 Lesson 3.1C: Constructing Perpendicular Bisectors Lesson 3.1C Student Inquiry What exactly is a perpendicular bisector? This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 1 49

52 Student Inquiries How do I draw a line segment perpendicular to another line segment? BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: answer How do I describe why they are perpendicular? example How do I draw a perpendicular bisector of a line segment using different construction methods? answer example How can I verify my results? answer example 50 MATH 7 etext Open School BC Module 3, Section 1

53 Lesson 3.1C: Constructing Perpendicular Bisectors Lesson 3.1C Thinking Space Introduction Imagine that you and a friend are exploring in Egypt, searching for archeological treasure. You want to begin looking at the Great Pyramid and your friend wants to start searching at the Sphinx nearby. You want to find a meeting spot that is the same distance from the Pyramid and the Sphinx to meet for lunch. If you have a map and geometry tools, you can find the meeting spot on your map. Let find s out how! Explore Online Looking for more practice or just want to play some fun games? If you have internet access, go to the Math 7 website at: Look for Lesson 3.1C: Constructing Perpendicular Bisectors and check out some of the links! Open School BC MATH 7 etext Module 3, Section 1 51

54 Warm-up In Lesson 3.1A, we talked about right angles and perpendicular lines. Look back to Lesson 3.1A, to review perpendicular lines. Perpendicular line segments are indicated by the symbol: For example, AB CD. C Thinking Space A B Recall that the symbol for a 90 angle or right angle is indicated by the symbol: D In Lesson 3.1B, you learned that a perpendicular bisector is a line that intersects a line segment at 90 and divides it into two equal lengths. For example, the letter T is a line segment with a perpendicular bisector. 52 MATH 7 etext Open School BC Module 3, Section 1

55 Explore The Greek word bisector means to cut into two equal parts (bi = two and sector = cutting or dividing into parts). Lesson 3.1C Thinking Space! There are many uses for perpendicular bisectors. You can draw a perpendicular bisector on a map to help you measure equal distances from two points. Perpendicular bisectors are very important to designers. For example, you can design a kitchen table or bridge and construct a perpendicular bisector that would act as the support post. Can you think of any other examples? Let s find out how to construct perpendicular lines and bisectors using a variety of methods. Constructing Perpendicular Bisectors Method 1: Folding Paper Construction Step 1: Mark a point A on your page representing the pyramid. Mark a point B to represent the Sphinx.? Why does gravity make a perpendicular bisector important in building construction? How can we find the perpendicular bisector? Step 2: Draw a line segment between A and B with your ruler. A B Open School BC MATH 7 etext Module 3, Section 1 53

56 Step 3: Fold the map so that A is directly on top of B. Thinking Space A B Step 4: Unfold the map and draw a line along the crease. This line is the perpendicular bisector. It forms a right angle with line segment AB. perpendicular bisector C 4.5 cm 4.5 cm A fold line B There are two ways to check your answer: Draw a point on the perpendicular bisector. This point should be the same distance from A and B. Measure the line segment with your ruler. The perpendicular bisector should divide the line in half. Use your protractor to measure the angle between the perpendicular bisector and line segment. It should equal MATH 7 etext Open School BC Module 3, Section 1

57 Method 2: Compass and Straight Edge Construction Lesson 3.1C Step 1: Mark a point A on your page representing the pyramid. Mark a point B to represent the Sphinx. Thinking Space Step 2: With your ruler, draw a line segment from A to B. A B Step 3: Adjust your compass so that its radius is more than half of the distance between A and B. Tighten the screw so that the distance between the point and the pencil does not change. A B Step 4: With the compass point at A, draw an arc between A and B. Step 5: Don t adjust your compass. With the compass point at B, draw another arc between A and B. Open School BC MATH 7 etext Module 3, Section 1 55

58 Step 6: The shape you have drawn is called a vesica. Label the top C and the bottom D. C and D are the two places where the arcs intersect. Step 7: Use a ruler to draw a line through C and D. Thinking Space perpendicular bisector C A B D 4.5 cm 4.5 cm A 2.2 cm 2.2 cm B Line CD is perpendicular to line AB. How can you check this? Measure the angle with your protractor. It should be 90. Line CD bisects line AB. How can you check this? Measure the distance from A to the point of intersection. Measure the distance from B to the point of intersection. These measurements should be the same. Line CD is the perpendicular bisector of CD. 56

59 Method 3: Protractor and Ruler Construction Lesson 3.1C Step 1: Step 2: Mark a point A on your page representing the pyramid. Mark a point B to represent the Sphinx. Draw a line segment between A and B with your ruler. Thinking Space A B Step 3: Step 4: Measure the line segment with your ruler and mark the centre. Label the point C. Place your protractor on the line segment AB, matching the vertex with point C. Step 5: Make a mark with your pencil at A C B

60 Step 6: Draw a line from point C to the 90 mark. This is the perpendicular bisector. Thinking Space perpendicular bisector D 4 cm 4 cm A C 2.2 cm 2.2 cm B Remember: There are two ways to check your answer. Try them both! Method 4: Right Triangle and Ruler Step 1: Step 2: Mark a point A on your page representing the pyramid. Mark a point B to represent the Sphinx. Draw a line segment between A and B with your ruler. A B 58 MATH 7 etext Open School BC Module 3, Section 1

61 Step 3: Measure the line segment AB with your ruler and mark the centre. Step 4: Label the centre Point C. Lesson 3.1C Thinking Space Step 5: Step 6: Place your right triangle on the line segment AB. Draw a line along the edge of your triangle starting at Point C. This is the perpendicular bisector. D A C B perpendicular bisector D 5 cm 5 cm A 2.2 cm 2.2 cm C B What is your favourite way to check your answer? Use it to make sure the line you drew really is a perpendicular bisector. 59

62 Practice 1. Describe what perpendicular lines are. 2. What is the symbol for perpendicular lines? 3. Describe what the perpendicular bisector of a line segment is. 4. Draw a perpendicular bisector of a line segment using different construction methods. First, draw a line segment that is the indicated length. Then construct the perpendicular bisector using the given construction method. a. 7 cm compass and straight edge (ruler) 60 MATH 7 etext Open School BC Module 3, Section 1

63 b. 10 cm protractor and ruler Lesson 3.1C c. 17 cm right angle and ruler 5. Can you make sure that the perpendicular bisector is correct? Open School BC MATH 7 etext Module 3, Section 1 61

64 6. This sign needs a post to support it. Draw where the perpendicular bisector support should be placed. BEWARE of Bears 62

65 7. Dan and Megan decide to visit a zoo. They enter the zoo at different gates and want to meet up at a location that is close to the same distance from each gate. Which would be a better exhibit to meet at the Zebras, the Monkeys or the Elephants? Lesson 3.1C Gate B Gate A Turn to the Answer Key at the end of the Module and mark your answers. 63

66 64 MATH 7 etext Open School BC Module 3, Section 1

67 Lesson 3.1D: Constructing Angle Bisectors Lesson 3.1D Student Inquiry How do I draw the bisector of an angle? This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 1 65

68 Student Inquiries How do I draw the bisector of a given angle? Is there more than one method that I could use? How do I make sure that the angle bisector is correct? BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: answer example answer example 66 MATH 7 etext Open School BC Module 3, Section 1

69 Lesson 3.1D: Constructing Angle Bisectors Introduction You and your friend are planning a puppet show. You are in charge of building the puppet stage tent for the show. Your stage frame is finished, you ve nailed the curtain material to cover the entrance. All is going well, but then you realize that there is no opening in the front of the tent to perform your show and the tent is already finished! What are you going to do? Lesson 3.1D Thinking Space The entrance of the tent is in the shape of an equilateral triangle (all sides are equal). You will need to cut the material right down the middle of the triangle so that each flap is equal in size and can open to the side. In this lesson, you will learn how to cut angles in half, so that each side is equal. Explore Online Looking for more practice or just want to play some fun games? If you have internet access, go to the Math 7 website at: Look for Lesson 3.1D: Constructing Angle Bisectors and check out some of the links! Open School BC MATH 7 etext Module 3, Section 1 67

70 Warm-up Before you begin this lesson, you may find it helpful to review Lesson 3.1A and how to use your protractor to measure angles. You may also find it helpful to review the terminology you learned in Lesson 3.1B. Thinking Space! Words such as angle bisector and perpendicular bisector seem a bit overwhelming at first. Don t worry, their meanings will make more sense as you work with them and learn how to construct these shapes. 68 MATH 7 etext Open School BC Module 3, Section 1

71 Explore Remember from the last lesson that the Greek word bisector means to cut into two equal parts (bi = two and sector = cutting or dividing into parts). Angle bisector means to cut an angle into two equal angles. Lesson 3.1D Thinking Space! Let s find out what an angle bisector looks like. Step 1: Step 2: Draw a right angle using your right triangle on paper. Label the points on the angle with W, X, Y, like the one shown below. W X Y Step 3: Fold the piece of paper so that XY lies on top of WX. W Y X Fold paper over Step 4: Make a point on the fold crease and label it Z. Step 5: Draw a line segment from X to Z on the fold crease. Line segment XZ is an angle bisector for WXY. W Z Fold line X Y Open School BC MATH 7 etext Module 3, Section 1 69

72 Use a protractor to measure the two angles: WXZ and ZXY. What do you notice about the two angles? Thinking Space WXZ = 45 ZXY = 45 What can you conclude about an angle bisector? How To Draw an Angle Bisector: You are colouring a chalk drawing in the park for a birthday party. You need to draw an arrow towards the picnic area in the park so that people know where to go. Using a right triangle, you can draw the arrowhead. How do you draw the angle bisector of the arrowhead to complete the arrow? BIRTHDAY PARTY THIS WAY You can construct angle bisectors using a variety of tool combinations and methods. 70 MATH 7 etext Open School BC Module 3, Section 1

73 Method 1: Draw an Angle Bisector with a Compass and Ruler Lesson 3.1D Step 1: Step 2: Draw an angle and label it ABC. Place your compass point on B and draw an arc that intersects AB and CB. Thinking Space Step 3: Label the intersection points D and E. A D B BIRTHDAY PARTY THIS WAY E C Step 4: Step 5: Step 6: Place your compass on point D and draw an arc within the angle. Without changing the radius of the compass, place your compass on point E and draw another arc within the angle. Draw a line starting at B going through the intersection point of the two arcs. This is the angle bisector. angle bisector A D B BIRTHDAY PARTY THIS WAY E C Open School BC MATH 7 etext Module 3, Section 1 71

74 How to check: Use you protractor to measure the two angles. Are they the same? Thinking Space angle bisector BIRTHDAY PARTY THIS WAY A D E C B Note: If you placed a transparent mirror on the angle bisector line, Point D and line segment AB will be a reflection of Point E and line segment CB. These two points and angles are mirror images of each other. Method 2: Draw an Angle Bisector with a Protractor and Ruler Step 1: Step 2: Step 3: Draw an angle and label it ABC. Measure angle ABC. Write down the angle and divide it in half. A B BIRTHDAY PARTY THIS WAY C Step 4: Place the base of your protractor on line segment CB. 72 MATH 7 etext Open School BC Module 3, Section 1

75 Step 5: Centre the protractor on point B. Lesson 3.1D Step 6: Make a point D at the size of the angle you found in Step 3. (In this example, our angle is 45 ). Thinking Space A D B BIRTHDAY PARTY THIS WAY C Step 7: Use a ruler to draw a line from B to D. Step 8: Measure the two angles with your protractor to confirm they are equal. D BIRTHDAY PARTY THIS WAY A C B Step 9: Line segment DB is the angle bisector. Which method do you prefer the compass and ruler, or protractor and ruler? Why? Open School BC MATH 7 etext Module 3, Section 1 73

76 Practice 1. Construct the angle bisector in the following angles. a. b. c. d. e. 2. Draw a 60 angle with a square triangle. Construct the angle bisector using a compass and ruler. 74 MATH 7 etext Open School BC Module 3, Section 1

77 3. Use a protractor to draw each angle. Then construct the angle bisector. Use a different method of construction for each angle. Lesson 3.1D a. 130 angle compass and ruler b. 70 angle protractor and ruler Open School BC MATH 7 etext Module 3, Section 1 75

78 4. Draw two triangles one triangle and one 45 triangle. Using the method that you prefer, draw angle bisectors for each angle in the two triangles. What do you notice about the bisectors? Turn to the Answer Key at the end of the Module and mark your answers. 76 MATH 7 etext Open School BC Module 3, Section 1

79 Lesson 3.1E: Constructing Parallel Line Segments Lesson 3.1E Student Inquiry How do I draw parallel lines? This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 1 77

80 Student Inquiries How do I draw a line segment that is parallel to another line segment? BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: answer example Why are the two lines parallel? answer example 78 MATH 7 etext Open School BC Module 3, Section 1

81 Lesson 3.1E: Constructing Parallel Line Segments Lesson 3.1E Thinking Space Introduction Streets and avenues are a familiar way to remember the definition of parallel and perpendicular lines. Many cities are divided into East-West Avenues and North-South Streets (or vice versa). Looking at these cities from a birds eye view, streets and avenues will form a grid of parallel lines and perpendicular lines. For example, Main Street and 1st Street will never intersect but run in the same direction. They are parallel to each other. Second Street and 1st Avenue will cross each other at a right angle (90 angle) and create a traffic intersection. Main St 1st St 2nd St 3rd St 2nd Avenue 1st Avenue What shape is created at the intersection of two streets and an avenue? If you said a square, you re correct! A kitchen table top runs parallel to the floor. Think of what would happen if the table surface was not parallel with the floor. The food would probably fall off the table onto the floor! The ability to draw parallel lines is a very important skill to have for engineers, carpenters, and architects. Let s learn how to draw parallel lines! Explore Online Looking for more practice or just want to play some fun games? If you have internet access, go to the Math 7 website at: Look for Lesson 3.1E: Constructing Parallel Line Segments and check out some of the links! Open School BC MATH 7 etext Module 3, Section 1 79

82 Warm-up You may find it helpful to review Lesson 3.1A and the definitions of parallel lines and line segments. Thinking Space A line segment is a straight line between two points. It has two definite end points and a definite length. A B Parallel lines never meet. The two lines are always equally distant from each other. A C 1 cm 1 cm B D Parallel line segments are indicated by the symbol: For example, this means line segment AB is parallel to line segment CD. AB CD 80 MATH 7 etext Open School BC Module 3, Section 1

83 Explore Which of the following line pairs are parallel? Even if two lines look parallel, they might not be looks can be deceiving. How can you show that two lines are parallel? Lesson 3.1E Thinking Space a. b. c. d. Constructing Parallel Line Segments Method 1: Right Triangle and Ruler Step 1: Step 2: Step 3: Use a ruler to draw a line segment. Label the endpoints A and B. Line up the longer edge of a 30, 60 right triangle along the line segment AB. Hold a ruler along the edge of the triangle. A B Open School BC MATH 7 etext Module 3, Section 1 81

84 Step 4: Slide the triangle downwards or upwards along the ruler. Thinking Space A B Step 5: Draw a line segment parallel to AB. A B C D Step 6: Check your work. Measure the distance between the parallel line segments at two locations. If the two distances are equal in length, the line segments are parallel. A B AB CD 2.5 cm 2.5 cm C D 82 MATH 7 etext Open School BC Module 3, Section 1

85 Method 2: Compass and Ruler Step 1: Use a ruler and draw a line segment (eg. AB ). Lesson 3.1E Thinking Space A B Step 2: Step 3: Construct a perpendicular bisector on the line segment. Label two points (eg. C, D) on one arm of the perpendicular bisector. D We constructed perpendicular bisectors in Lesson 3.1C. A C B Step 4: Using these two points (C, D), construct the perpendicular bisector of this first perpendicular bisector. Label the second perpendicular bisector, EF. E F A C B Line segment EF is parallel to line segment AB. AB EF E F A B Open School BC MATH 7 etext Module 3, Section 1 83

86 Method 3: Protractor and Ruler Step 1: Step 2: Step 3: Draw a line segment with a ruler. W X Use your protractor to draw a 90 angle at both points W and X. Use your ruler to draw equal length perpendicular line segments from points W and X. Label the end points of each line segment (eg. Z, Y). Z Y Thinking Space 3 cm 3 cm W X Step 4: Connect the perpendicular line segments with a line to create a rectangle. Z Y 3 cm 3 cm W X The top and bottom line segments ( WX and ZY ) are parallel. The two side line segments ( ZWand YX ) are also parallel. 84 MATH 7 etext Open School BC Module 3, Section 1

87 Practice Lesson 3.1E 1. Which of the following line segments are parallel? Use your ruler to check your answer. a. b. c. d. 2. Draw two parallel line segments that are each 5 cm long using the compass and ruler method. Open School BC MATH 7 etext Module 3, Section 1 85

88 3. Draw a rectangle with length (6 cm) and width (3 cm) using the protractor and ruler method. 4. Draw two parallel line segments that are 4 cm apart using the Right Triangle and Ruler method. Turn to the Answer Key at the end of the Module and mark your answers. 86 MATH 7 etext Open School BC Module 3, Section 1

89 Section Summary Summary 3.1 Line Segments A line segment is a line between two given points. It has finite or defined length. Examples: This is a line segment: A B This is a line (and not a line segment): A B This is a ray: A line and a ray have infinite or undefined length. A Measure an Angle Using a Ruler and Protractor Parallel Line Segments These two line segments are parallel: A B C D AB CD Open School BC MATH 7 etext Module 3, Section 1 87

90 Perpendicular Line Segments These two line segments are perpendicular: C 90 A B How to Construct Perpendicular Bisectors Method 1. Folding Paper A B Method 2. Compass and Straight Edge Use a compass to draw intersecting circles from two points on a line segment. Draw a line through the points of intersection. This line is the perpendicular bisector. A B 88 MATH 7 etext Open School BC Module 3, Section 1

91 Method 3. Protractor and Ruler Summary 3.1 Use a ruler to measure the midpoint on a line segment. Then use a protractor to draw a 90 angle at the midpoint. How to Construct Angle Bisectors Method 1. Folding paper to Construct an Angle Bisector Fold the line segments of an angle on top of each other. The fold-line is the angle bisector and it divides the angle in two. W Y X Fold paper over W Z X Y Fold line Method 2. Using a Compass and Ruler to Construct an Angle Bisector Open School BC MATH 7 etext Module 3, Section 1 89

92 Method 3. Using a Protractor and Ruler to Construct an Angle Bisector How to Construct Parallel Line Segments Method 1. Using a Right Triangle and Ruler to Construct Parallel Line Segments A B C D Method 2. Using a Compass and Ruler to Construct Parallel Line Segments Construct the perpendicular bisector of a line segment. Then construct the perpendicular bisector of the first perpendicular bisector. E F A C B Method 3. Using a Protractor and Ruler to Construct Parallel Line Segments Z Y 3 cm 3 cm W X 90 MATH 7 etext Open School BC Module 3, Section 1

93 Section Challenge Challenge 3.1 Sarah and Tom want you to design a tree house for their backyard. It sounds very exciting and you have lots of ideas. Quickly, you realize that there are many things that you will need to consider. To start, you ll have to choose a suitable tree. You ll also need to know how to draw parallel and perpendicular line segments. The floor of the tree house has to run parallel to the ground so that people can stand up straight. Also, the walls in the tree house have to run perpendicular to the floor, so that the walls won t fall over. You ll need to design and build a ladder so that they can climb up to their tree house. Sarah and Tom s friends also want you to draw a map with an arrow pointing out the direction to the tree house, so they know how to find it. Instructions are provided for you here, as well as space to complete your work. Feel free to use your own paper if you need more space. Note: If you want, review the Section Challenge at the beginning of the section. 1. On the site plan, draw line segments from the back door of Sarah and Tom shouse to each of the trees in the yard. 2. Measure each line segment. 3. Choose the tree that is closest to the back door. Oak Tree Apple Tree Back Door Front Door Cherry Tree Now you need to design the tree house. Draw your design on the tree image on the next page. 4. Construct a parallel line segment between the ground and half way up the tree. This is the floor of the tree house, which needs to run parallel to the ground. 5. Construct perpendicular lines going upwards from both ends of the floor to make walls. Open School BC MATH 7 etext Module 3, Section 1 91

94 6. Design the tree house however you like. You should include a door, windows, and a 30 sloping roof. 7. Construct a perpendicular bisector from the bottom of the tree house to the ground, as a support post. 8. Construct a ladder going from the ground to the tree house. You need to construct parallel line segments to make the rungs of the ladder. 9. Add any other details you would like to complete the design of a tree house. 10. Sarah and Tom s friends will need a map with an arrow pointing out the direction to the tree house, so they know how to find it. On the top view map above, construct an arrow pointing out the direction to the tree house. You need to draw an angle for the head of the arrow and an angle bisector as the tail of the arrow. 92 MATH 7 etext Open School BC Module 3, Section 1

95 Section 3.2: Areas of Polygons 2 Contents at a Glance Pretest 95 Section Challenge 98 Lesson A: How Do We Measure Things? 99 Lesson B: Finding the Area of Triangles 117 Lesson C: Finding the Area of Parallelograms 131 Lesson D: Problem Solving with Area 145 Section Summary 159 Learning Outcomes By the end of this section you will be better able to: describe the metric units of measurement. define the term polygon and identify examples of polygons. develop and use a formula to find the area of a triangle. develop and use a formula to find the area of a parallelogram. solve problems that involve finding the area of triangles and parallelograms. Open School BC MATH 7 etext Module 3, Section 2 93

96 94 MATH 7 etext Open School BC Module 3, Section 2

97 Pretest 3.2 Pretest 3.2 Complete this pretest if you think that you already have a strong grasp of the topics and concepts covered in this section. Mark your answers using the key found at the end of the module. If you get all the answers correct (100%), you may decide that you can omit the lesson activities. 1. Match the unit that you would use to measure the following items. distance between a. mm Vancouver and Kamloops your own height b. cm length of a spider c. m height of your closet d. km 2. Give the symbol for each of the following: a. three square metres b. five cubic metres c. one square centimetre d. two millimetres Open School BC MATH 7 etext Module 3, Section 2 95

98 3. Fill in the blanks. a. 10 millimetres (mm) = centimetre (cm) b. 10 cm = decimetre (dm) c. cm = 1 metre (m) d. m = 1 kilometre (km) 4. Find the volume of a box with a length of 4 cm, a width of 3 cm, and a height of 1.5 cm. Draw a diagram and label it. 5. A triangle has a base of 6 m. The height is 4 m. Find the area of the triangle. 96 MATH 7 etext Open School BC Module 3, Section 2

99 6. Complete the following table. You are finding the base, height, or area for a parallelogram. Round your answer to two decimal places, if needed. Pretest 3.2 BASE HEIGHT AREA 125 m 25 m 36 cm 18 cm 10.2 m 8.4 m 0.25 m 4.93 cm 7 cm 49 cm² 3.6 m 7.2 m² 7. Robyn has constructed a garden in the shape of a triangle. She wants to plant as many daffodils as she can. The base of the garden is 240 cm and the height is 3.5 m. She knows that she can plant 35 daffodils per square metre of garden. How many daffodils can she plant? Turn to the Answer Key at the end of the Module and mark your answers. Open School BC MATH 7 etext Module 3, Section 2 97

100 Section Challenge The local recycling centre has asked you to design a poster for them. Part of the challenge is to use the least amount of paper. The recycling centre has a saying, less is more! They would like the poster to be in the shape of one of the triangles or parallelograms shown below. Can you choose the shape that would use the least amount of paper? In this section, you will learn how to find the area of triangles and parallelograms. a. b. 14 cm 0.12 m 16 cm 15 cm c. d. 18 cm 0.14 m 13 cm 8 cm e. 0.4 m 25 cm If you re not sure how to solve the problem now, don t worry. You ll learn all the skills you need to solve the problem in this section. Give it a try now, or wait until the end of the section it s up to you! 98 MATH 7 etext Open School BC Module 3, Section 2

101 Lesson 3.2A: How Do We Measure Things? Challenge Lesson 3.2A Student Inquiry Perimeter, area and volume! This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 2 99

102 Student Inquiries What are the metric units of measurement? BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: answer example What is a polygon? answer example What is perimeter and how do I measure it? answer example What is area and how do I measure it? answer example What is volume and how do I measure it? answer example 100 MATH 7 etext Open School BC Module 3, Section 2

103 Lesson 3.2A: How Do We Measure Things? Introduction The word geometry comes from the Ancient Greek word geometria and means to measure the Earth. Lesson 3.2A Thinking Space! Geometry was originally developed and used to measure common features of Earth, such as farmland. Every year in Ancient Egypt, the Nile River would overflow and wash away the markers that divided one farmer s field from another. Land surveyors used geometry to measure the fields so that farmers had equal sized farms. They marked off the land into shapes such as squares and triangles, and then calculated the area. By calculating the area, they could ensure that each farmer s field covered the same amount of land space. In this lesson, you are going to review the different tools and units used to measure things. This is a quick lesson, but it s an important review. Soon, you too will be able to calculate the land area of a farmers field, like the Ancient Egyptians. Good luck! Explore Online Looking for more practice or just want to play some fun games? If you have internet access, go to the Math 7 website at: Look for Lesson 3.2A: How Do We Measure Things? and check out some of the links! Open School BC MATH 7 etext Module 3, Section 2 101

104 Warm-up What tools and units would you use to measure the length of a stamp, width of a shoebox, or length of a soccer field? Thinking Space Can you use the same measuring tool in each situation? You could use a ruler to measure the stamp and shoebox, but probably not to measure a soccer field. Can you use the same unit of measurement? You could use centimetres to measure the stamp and shoebox, but probably not for the soccer field. You would probably use metres to measure the soccer field. Let s review the metric units of measurement. Length/Width 10 millimetre (mm) = 1 centimetre (cm) 10 cm = 1 decimetre (dm) 100 cm = 1 metre (m) 1000 m = 1 kilometre (km) Unit Conversions Can you convert one metric unit of measurement into another metric unit of measurement? Here are some examples for you to review. How many metres are equal to 125 cm? 100 cm = 1 m You divide 125 cm by 100 = 1.25 m How many centimetres are equal to 0.34 metres? 1 m = 100 cm You multiply 0.34 m by 100 cm = 34 cm 102 MATH 7 etext Open School BC Module 3, Section 2

105 Here s a method you can use to visualize conversions: Lesson 3.2A km m dm cm mm km to m Add 3 zeros 1 km = 1000 m m cm mm m to cm Add 2 zeros 1 m = 100 cm m cm cm to m Move the decimal point 2 spaces 1.0 cm = 0.01 m Tools of Measurement Ruler Most are 30 cm in length or 300 mm. Metre stick 100 cm 1 m in length (or 100 cm). Tape measure Rope or chain Most are marked in centimetres and inches. Used to measure distances in and around the house (eg. walls and floors). Knots in the rope or links in the chain are a given distance from one another. (eg. length between knots = 1 m. 5 knots in length is = 5 m) Open School BC MATH 7 etext Module 3, Section 2 103

106 Trundle wheel Used to measure long distances. It is made up of a wheel, a handle, and a clicking device that clicks after one revolution of the wheel. The circumference of the wheel (distance around) is exactly 1 metre. One revolution of the wheel equals 1 metre distance traveled on the ground. Every time the wheel makes a rotation, a click is heard and counted. The total number of clicks counted equals the number of metres traveled. Protractor Used to measure angles (0 to 180 ). Compass Can be used to measure distances of line segments. Pencil Used to mark points or line segments. Straight edge Used to draw a line between points which is then measured. Chalk line The chalk string is stretched between two points. The string is snapped in the middle, leaving a chalk line that is then measured. 104 MATH 7 etext Open School BC Module 3, Section 2

107 Explore In this section, you will be learning how to measure shapes known as polygons. The Greek word polygon means many angles. A polygon is a shape with straight sides. Polygons are named according to the number of sides and angles they have. For example, a triangle has three sides and three angles. Two other familiar polygons are the square and rectangle. Lesson 3.2A Thinking Space! Here are some examples of polygons: These are not polygons. Why do you think they are not polygons? Unlike polygons, these shapes are made up of curves, not just straight line segments. Measuring Line Segments, Polygons, and Solid Objects 1 m 1 m Metre stick 1 m 1 m 1 m 1 m LENGTH AREA VOLUME Open School BC MATH 7 etext Module 3, Section 2 105

108 Line segments are one-dimensional, which means you can measure the length of a line segment with a ruler. Plane figure polygons are two-dimensional and you can measure the perimeter (using length of sides), but must calculate the area (using length and width). Thinking Space Solid objects are three-dimensional and you can measure their perimeter, and calculate their surface area and their volume (using length, width, and height). Perimeter of Common Polygons Triangles and Rectangles The perimeter is the distance around the polygon or sum of the lengths of the sides. We measure perimeter in the same units as specified on the figure. For example, if the length of the side is measured in cm, the perimeter will be in centimetres. Let s look at two examples: Triangles a b c Perimeter of a triangle = a + b + c c = 5 b = 4 a = 3 P = a + b + c P = P = 12 The perimeter of ABC is 12 cm. 106 MATH 7 etext Open School BC Module 3, Section 2

109 Rectangles l stands for the length of the rectangle and w stands for the width. Lesson 3.2A Thinking Space l // w / / w // l Perimeter of a rectangle = l + w + l + w (You have 2 ls and 2 ws) P = 2l + 2w 1.2 m // / / 80 cm // What do you notice about the metric units of the length and width in this figure? Before the perimeter can be calculated, the length of each side must be in the same metric unit. Here, we must change both length and width to metres, or both to centimetres. In centimetres: l = 1.2 m = 120 cm w = 80 cm P = 2l + 2w P =(2 120) + (2 80) P = P = 400 cm The perimeter is 400 cm OR 4 m. In metres: l = 1.2 m w = 80 cm 80 cm 100 = 0.8 m P = 2l + 2w P = (2 1.2) + (2 0.8) P = P = 4 m Open School BC MATH 7 etext Module 3, Section 2 107

110 Metric Units of Area Area is the amount of surface within a shape. Do you remember how to measure the area within the rectangle? Thinking Space You ll now have to consider two dimensions of the figure to find area: length and width. Perimeter is measured in linear units such as centimetres or metres. Do you think the metric unit used to measure area will be different from the linear units used to measure perimeter? We measure area in square units such as square metres (m 2 ) and square centimetres (cm 2 ). You can think of area as the number of square units that covers a closed figure.! 1 m 1 m The standard metric unit of area is the square metre. This is the area contained in a square that is 1 metre by 1 metre (or m m). Using what we know about exponents, m m can also be written as m 2. We say 1 square metre when we see the unit 1 m 2. What unit would you use to measure large distances, like the distance between Vancouver and Calgary? What unit would you use to measure small distances, like the distance between the wings of a butterfly? You can also say one metre squared. To measure large areas of land, the square kilometre is used. To measure small areas, you might use the square millimetre. Remember: Measurements of area must always include a unit of measurement. Your answer will not be complete if you do not include the unit of measurement. For example, 5 does not mean the same as 5 m or 5 m 2. Always be sure to include a unit of measurement whenever you are recording linear, area, or volume measurements. 1 mm² 1 square millimetre 1 cm² 1 square centimetre 1 dm² 1 square decimetre 1 m² 1 square metre 1 km² 1 square kilometre 108 MATH 7 etext Open School BC Module 3, Section 2

111 Area of Rectangles Lesson 3.2A l // Thinking Space w / / w // l Area of a rectangle = l w A = lw 4 cm 2 cm A = lw A = 4 2 A = 8 square centimetres or 8 cm² The area is 8 cm². One centimetre is equal to one hundredth of a metre. It is important that you do not treat 1 cm 2 as being equal to one hundredth of a square metre. Let us see why this is so. Suppose you want to calculate the area of a piece of carpeting that is 1 metre by 1 metre. Area = length width A = 1 m 1 m A = 1 m 2 The area of the carpet is 1 m 2. Since 1 metre = 100 centimetres, you can also calculate the area of the piece of carpet in square centimetres. A = l w A = A = cm 2 The area of the carpet is cm 2. Therefore, 1 square metre is made up of square centimetres. Open School BC MATH 7 etext Module 3, Section 2 109

112 Metric Units of Volume Volume is the amount of space occupied by a 3-dimensional (solid) object. Volume is measured in cubic units. Thinking Space The standard metric unit of volume is the cubic metre. This is the space contained in a cube that is 1 metre long, 1 metre wide, and 1 metre high. 1 m 1 m 1 m The symbol for cubic metre is 1 m³. If length, width, and height are measured in centimetres, the volume is measured in cubic centimetres. The symbol for cubic centimetres is 1 cm³. One cubic centimetre = 1 cm 1 cm 1 cm = 1 cm³ One cubic millimetre = 1 mm 1 mm 1 mm = 1 mm³ Volume of a box = length width height How many cubes measuring 1 cm by 1 cm by 1 cm (1 cm³) do you think will fit in this box? 3 cm 1 cm 1 cm 1 cm 3 cm 3 cm 110 MATH 7 etext Open School BC Module 3, Section 2

113 Volume = 3 cm 3 cm 3 cm V = 27 cm³ You can fit 27 cubes measuring 1 cm by 1cm by 1cm (1 cm³) into the box. Lesson 3.2A Thinking Space Here s a quick reference for measuring length, area, and volume: 1 m 1 m Metre stick 1 m 1 m 1 m 1 m LENGTH AREA VOLUME The metre stick is 1 metre in length. One dimension: length The square is 1 metre by 1 metre, or 1 square metre in area. Two dimensions: length and width The cube is 1 metre by 1 metre by 1 metre, or 1 cubic metre in volume. Three dimensions: length and width and height Open School BC MATH 7 etext Module 3, Section 2 111

114 Practice 1. Match each tool to the situation where you would use the tool. used to measure the length a. Ruler of your living room used to measure the length b. Metre stick around a garbage can used to measure the length c. Tape measure around a park used to measure the angle d. Rope or chain between the hands of a clock pointing at 3 o clock used to measure the height e. Trundle wheel of your Science Fair backboard used to measure the length f. Protractor of a line segment 2. Match the unit that you would use to measure the following items: distance between a. mm Vancouver and Calgary your own height b. cm length of a ladybug c. m height of your closet d. km 112 MATH 7 etext Open School BC Module 3, Section 2

115 3. Give the symbol for each of the following: Lesson 3.2A a. two square metres b. five cubic metres c. one square centimetre d. four millimetres e. one cubic millimetre f. three square kilometres g. one kilometre h. three cubic centimetres i. five square millimetres j. two centimetres Open School BC MATH 7 etext Module 3, Section 2 113

116 4. Find the perimeters of these polygons and include units: a. // // 5 cm // P = b. 12 cm // / / 4.6 cm // P = c. 1 m // // // 100 cm // P = 114 MATH 7 etext Open School BC Module 3, Section 2

117 5. Rectangles can have the same perimeter yet be different in area. Complete the table by finding the area of these rectangles. Note: Areas are in square metres or m². Lesson 3.2A a. b. / 7 m // // / 1 m / 6 m // // / 2 c. 5 m // d. 4 m // / / 3 m // // // // l w P A a. 7 m 1 m 16 m b. 6 m 2 m 16 m c. 5 m 3 m 16 m d. 4 m 4 m 16 m Open School BC MATH 7 etext Module 3, Section 2 115

118 6. Find the volume of this box and include units. 6.2 cm 4 cm 4 cm V = Turn to the Answer Key at the end of the Module and mark your answers. 116 MATH 7 etext Open School BC Module 3, Section 2

119 Lesson 3.2B: Finding the Area of Triangles Lesson 3.2B Student Inquiry There are so many different types of triangles! This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 2 117

120 Student Inquiries How can I use the area of a rectangle to find the area of a triangle? Can I create a formula to determine the area of triangles? BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: answer example answer example 118 MATH 7 etext Open School BC Module 3, Section 2

121 Lesson 3.2B: Finding the Area of Triangles Introduction Triangles are one of the most common polygons. Architects and engineers use the triangle in many of their designs because of the strong qualities of a triangle. Triangle shapes are much more difficult to distort than squares, rectangles, or circular shapes. You can easily distort a square box or a rubber ball with your hands. Just push the two corners together of a cardboard box and it flattens. Not so with a triangle. Try making a triangle with toothpicks and frozen peas. Gently press on the top of the triangle. Your triangle will hold its shape! Lesson 3.2B Thinking Space Just look around at things in your daily life and see if you can spot some triangles. Can you see a triangle at the top of a kite? In the sails of a sailboat? In the truss holding up a bridge or railroad over a river? Most houses have triangular rooftops so that rain and snow can easily slide off the top of the house onto the ground. Triangles are included in many familiar building designs such as the Eiffel Tower or Sydney Opera House, or in quilt designs such as the Pinwheel. In this lesson, you will learn how to create a formula for determining the area of triangles by folding a piece of paper in half. You ll need graph paper to complete the activities in this lesson. Open School BC MATH 7 etext Module 3, Section 2 119

122 Warm-up A triangle is a shape with three sides and three angles. Tri means three in Greek, so triangle simply means three angles. Connect any three points with line segments and you have a triangle. Just make sure that all three points aren t on the same line. Thinking Space! A C B Line segments AB, BC, and CA are the sides of the triangle. Points A, B, and C are each called a vertex of the triangle. Points A, B, and C are the vertices of the triangle. (Vertices is the plural of vertex.) We name a triangle by naming its vertices. We say triangle ABC, and we write ABC. ALWAYS label vertices with capital letters. Try This! Step 1: Step 2: Step 3: Step 4: Step 5: On a piece of paper, draw three different shaped triangles. (They can be any size.) Cut out the three triangles. Tear off the three corners of one of the triangles. Place the corners on a protractor to measure the total angle of all three corners. Repeat Steps 3 4 for the two other triangles. What did you notice about the three corners of each triangle? 120 MATH 7 etext Open School BC Module 3, Section 2

123 Types of Triangles Is it possible to have a triangle with three 90 angles? The mathematician Roger Penrose was the first person to draw a particularly tricky triangle on paper. It s called the Penrose Triangle. Each angle in the Penrose Triangle looks like it is a right angle. Why is it impossible to make a real 3-Dimensional triangle like this? Lesson 3.2B Thinking Space Try making your own Penrose triangle with three pencils. You ll quickly see it s impossible! All angles in a triangle must add up to 180. The angles in a Penrose Triangle add up to 270, so it is not a real triangle, and only an illusion drawn on paper. There are many different types of triangles, as you will see below. However, all triangles have one thing in common: the angles of a triangle always add up to 180. Triangles can have names related to their sides: NAME EXAMPLE DEFINITION Equilaterial Triangle Three equal sides Three equal angles Each angle is 60 // // // Isosceles Triangle Two equal sides Two equal angles // // Scalene Triangle / // /// No equal sides No equal angles Open School BC MATH 7 etext Module 3, Section 2 121

124 Triangles can also have names that are related to the angles inside: NAME EXAMPLE DEFINITION Acute Triangle All angles are less than 90 Thinking Space Right Triangle Has a right angle (90 ) Obtuse Triangle Has an angle more than MATH 7 etext Open School BC Module 3, Section 2

125 Explore It s important that you start using the word base for length and height for width when you re working with triangles. When you re measuring the length and width of a triangle, the length is called base and the width is called height. Lesson 3.2B Thinking Space! length = base width = height Use the Area of a Rectangle to Find the Area of a Triangle You can use the area of a rectangle to find the area of a triangle. Let s find out how by following the steps: Step 1: Step 2: Step 3: On graph paper, draw two different sized rectangles with your pencil and ruler. Use your scissors to cut out the two rectangles. Figure out the length of the base and height for each rectangle. You can do this by counting how many squares are along the base and height. height base Step 4: Complete the table below using the dimensions from your cut out rectangles. (The first row is filled out, using the example above.) BASE HEIGHT AREA 9 units 5 units A = b h A = 9 5 A = 45 units² units units units² Step 5: Check the area by counting the number of squares in each rectangle. Open School BC MATH 7 etext Module 3, Section 2 123

126 Now let s create a formula for determining the area of triangles by using the method of folding a piece of paper in half. If you ve ever made a birthday card, yo know that you start by folding a piece of paper in half. Thinking Space If you unfold this card, you will see that you have created two identical rectangles, side by side. Now try folding one of your paper rectangles in half to create two identical triangles. b b h h h b You ll find that one rectangle can make two triangles. In the first table, you calculated the area of your paper rectangle. Can you guess the area of one of the triangles? Check your guess by counting the number of complete squares in the triangle. Remember, two half squares = 1 square. 124 MATH 7 etext Open School BC Module 3, Section 2

127 Create a Formula to Determine the Area of Triangles If the area of a rectangle is base height, then the area of the triangle must be half of that. You can calculate the area of a triangle by multiplying base height then dividing by 2. Lesson 3.2B Thinking Space h h b b h h b b h h b b h h h b b Area of a triangle = base multiplied by height divided by 2. The same equation can be written as base height. 2 Area of a triangle = bh 2 = bh 2 Open School BC MATH 7 etext Module 3, Section 2 125

128 Determining the Base and Height of a Triangle Before you can calculate the area of a triangle, you need to determine the base and height of a triangle. This takes some practice. Here are some steps to follow which will make it easier for you. Thinking Space Step 1: Step 2: Step 3: Decide which side of the triangle will be your base. The base can be any side of the triangle. Find the height of the triangle. Draw a perpendicular line from the base of the triangle to the vertex. With some triangles, the height of a triangle can be drawn outside the triangle. Calculate the area using the base and height: Area = ½bh Equilateral Triangle h Isosceles Triangle b h b Scalene Triangle h Right Triangle h b Acute Triangle h b Obtuse Triangle b b h 126 MATH 7 etext Open School BC Module 3, Section 2

129 Here are two examples of calculating the area of a triangle: 1. Find the area of the following triangle. Lesson 3.2B Thinking Space A = b h 2 A = A = 35 2 A = 17.5 units² 2. Find the area of the following triangle. 4 cm 6 cm A = b h 2 A = A = 24 2 A = 12 cm² Here s a tricky one: Can you take five toothpicks away from this figure to leave only five triangles? Open School BC MATH 7 etext Module 3, Section 2 127

130 Practice 1. Fill in the missing numbers. Base = units Height = units Base = units Height = units 2. Find the area of each triangle. Include the units. a. b. c. 11 units 14 units 8 cm 5.2 cm 30 mm 25 mm Area = Area = Area = 128 MATH 7 etext Open School BC Module 3, Section 2

131 3. Find the area of each triangle. Include the units. Hint: Make sure the units of the height and the base are the same before you find the area. Lesson 3.2B a. b. c. 3 cm 2 cm 9 units 2 cm 0.03 m 16 units Area = Area = Area = 4. A triangle has a base of 6 m. The height is 4 m. Find the area of the triangle. Open School BC MATH 7 etext Module 3, Section 2 129

132 5. If line segment BC is the base of ABC, use a dotted line to show the height of this obtuse triangle. Then, calculate the area of the triangle. Hint: The height can lie outside of the triangle A B C Turn to the Answer Key at the end of the Module and mark your answers. 130 MATH 7 etext Open School BC Module 3, Section 2

133 Lesson 3.2C: Finding the Area of Parallelograms Lesson 3.2C Student Inquiry Rectangles, squares, rhombus and parallelograms! This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 2 131

134 Student Inquiries How can I use the area of a rectangle to find the area of a parallelogram? Can I create a formula to determine the area of parallelograms? BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: answer example answer example 132 MATH 7 etext Open School BC Module 3, Section 2

135 Lesson 3.2C: Finding the Area of Parallelograms Lesson 3.2C Thinking Space Introduction Did you know that the square-shaped window in your house and the rectangular-shaped piece of paper in your book are both in the shape of a parallelogram? Parallelograms share similar properties to squares and rectangles. Build a rectangle with toothpicks and marshmallows. If you push down on the top, it will tilt. The shape is no longer that of a rectangle, but it s still a parallelogram. Does the area inside the toothpick rectangle change when you tilt it? In this lesson, you will learn how to use the area of a rectangle to find the area of all parallelograms. You ll need graph paper, scissors, pencil, and a ruler to complete the activities in this lesson. Open School BC MATH 7 etext Module 3, Section 2 133

136 Warm-up First, let s review how to find the area of rectangles from the previous lesson. Remember square units are used to measure area. Thinking Space In this rectangle, there are 4 groups of 2 square centimetres (4 2). You can find the area of this rectangle by multiplying base height. 4 cm 2 cm 4 groups of 2 squares 2 squares Area Area = 4 squares 2 squares = 8 square units = 8 units² = 4 cm 2 cm = 8 cm² 134 MATH 7 etext Open School BC Module 3, Section 2

137 Explore What is a Parallelogram? Compare these two figures. Lesson 3.2C Thinking Space Figure 1 Figure 2 How are they the same, and how are they different? Compare these two figures. Figure 3 Figure 4 How are they the same, and how are they different? All four figures are parallelograms, but have different names. Figure 1 is a rectangle. Figure 2 is a parallelogram. Figure 3 is a square. Figure 4 is a rhombus. Open School BC MATH 7 etext Module 3, Section 2 135

138 A parallelogram is a four-sided shape with opposite sides parallel and equal in length. Parallelograms with special names: Square all sides are equal and all angles are 90. Thinking Space! Rectangle opposite sides are equal and all angles are 90. Rhombus all sides are equal and opposite sides are parallel. Parallelograms and Rectangles To do this activity you will need: scissors graph paper pencil Step 1: Draw a rectangle on graph paper, then cut it out. Step 2: What is the area of your rectangle? There are two ways to determine the area. 1. You could count all of the squares to figure out the area. 2. You could also count the number of squares along the base of the rectangle and the number of square along its height. Multiply base height to get the area. Are both of your answers the same? height base 136 MATH 7 etext Open School BC Module 3, Section 2

139 Step 3: Draw a parallelogram on graph paper, then cut it out. Lesson 3.2C Thinking Space Step 4: Count the squares in your parallelogram to determine the area. Do you think your answer is accurate? What was your strategy for counting the pieces of squares? Step 5: Cut off the triangle at the left side of your parallelogram. Step 6: Move the triangle you cut off to the right side of your parallelogram. Now you have a rectangle! You know how to find the area of a rectangle. What is the area of this rectangle? Is your answer the same as your answer in Step 4? Repeat Steps 4, 5, and 6 a few more times with parallelograms of different sizes. Open School BC MATH 7 etext Module 3, Section 2 137

140 Create a Formula to Determine the Area of Parallelograms The formula you use to find the area of each rectangle is: Thinking Space Area of a rectangle = base height In your thinking space, can you write a formula to find the area of a parallelogram? Did you guess that the formula to find the area of a parallelogram is the same as the formula to find the area of rectangles? A of rectangle = b h b h A of parallelogram = b h h b Area of a parallelogram = base height Be Careful: The height is NOT the length of the slanted side! The height is where the dotted line would be if you cut of the end of the parallelogram to make a rectangle. height base b h I saw that symbol in Lesson 3.1A. It means perpendicular. The height is perpendicular to the base. h h b b 138 MATH 7 etext Open School BC Module 3, Section 2

141 Example 1: The parallelogram below has a base of 4 units and a height of 3 units. Calculate the area using the formula Lesson 3.2C Thinking Space Area = base height. h b Here s the answer: Area = 4 units 3 units Area = 12 square units The area of the parallelogram is 12 units². Example 2: Try this one. 4 cm 6 cm base = cm height = cm A A = b h = square cm Open School BC MATH 7 etext Module 3, Section 2 139

142 Example 3: Look at the following parallelograms. In your thinking space, determine the height and base of each parallelogram. Then find the area of each parallelogram. Thinking Space a. b. c. 2 cm 2.3 cm 170 cm 150 cm 7 cm 4 cm 2 m 1 cm Did you get these answers? Once you re comfortable with parallelograms, try the Practice. BASE HEIGHT AREA a. 4 cm 2 cm 8 cm² b. 1 cm 7 cm 7 cm² c. 2 m 150 cm 3 m² 140 MATH 7 etext Open School BC Module 3, Section 2

143 Practice Lesson 3.2C 1. Find the area of the parallelogram below: base = units height = units Area = b h = units units = units² 2. Find the area of the parallelograms below: 5 cm 2 4 cm 8 Area = b h Area = = cm cm = ² = cm² Open School BC MATH 7 etext Module 3, Section 2 141

144 3. Find the area of the parallelograms below: 8 cm 1.5 m 1 m 4.5 m 2 cm base = height = base = height = Area = Area = 32 cm 0.3 m 40 cm base = height = Area = 142 MATH 7 etext Open School BC Module 3, Section 2

145 4. Complete the following table. Round your answer when necessary. Lesson 3.2C BASE HEIGHT AREA 125 m 25 m 36 cm 18 cm 10.2 m 8.4 m 0.25 m 4.93 cm 7 cm 49 cm2 3.6 m 7.2 m2 Turn to the Answer Key at the end of the Module and mark your answers. Open School BC MATH 7 etext Module 3, Section 2 143

146 144 MATH 7 etext Open School BC Module 3, Section 2

147 Lesson 3.2D: Problem Solving with Area Lesson 3.2D Student Inquiry What are the problem solving steps? This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 2 145

148 Student Inquiries What kinds of problems involve finding the area of triangles? What kinds of problems involve finding the area of parallelograms? What strategies can I use to solve problems involving area? BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: answer example answer example answer example 146 MATH 7 etext Open School BC Module 3, Section 2

149 Lesson 3.2D: Problem Solving with Area Introduction The Ancient Egyptians built the Great Pyramid around 5000 years ago. The Egyptian builders didn t have the tools that we have today, yet they were still able to build one of the most spectacular buildings in the world. Engineers are still trying to figure out how they did it. Did you know that the builders oriented the pyramid so that the four corners point almost perfectly north, south, east, and west? Incredibly, they used around 2.3 million stone blocks, weighing an average of 2.5 to 15 tons each, to build the structure. Engineers estimate that the builders would have had to set a block in place every two and a half minutes in order to complete the massive structure. Lesson 3.2D Thinking Space To build the Great Pyramid, the Ancient Egyptians must have had a step-by-step plan figured out, and a system in which to carry out their plan. The pyramid is proof of how effective they were at solving math problems to meet their practical needs, such as surveying land and building temples. How do you think they were able to come up with such a plan and carry it out? In this lesson, you will learn the steps involved to understand a problem involving area, make a plan to solve it, and carry out the plan. Open School BC MATH 7 etext Module 3, Section 2 147

150 Warm-up Imagine you are asked to build a pyramid. What are some of the things you would have to think about in making up a plan? For example, can you start putting stones in place right away, or do you have to first think about what your pyramid should look like? Can you use metal or clay to build your pyramid? In your thinking space, write down some of the things you need to plan before you start actually building a pyramid. Thinking Space You can also use a web to brainstorm some ideas. Start by writing, Make a plan to build a pyramid in the centre of the web. What step would be first in your plan? Hint: you will certainly need to choose the length of the base of the pyramid. You can use different strategies to solve a math problem. It is more important to remember that math problems generally can t be solved in one step. You have to start at the beginning, use what you know, and work your way through steps to arrive at the answer. Here are some steps the Ancient Egyptians might have followed to build the Great Pyramid: 1. Thought about what they wanted to do. 2. Brainstormed ideas. 3. Designed a picture of a pyramid on papyrus paper, using tools such as a ruler and compass. 4. Figured out the dimensions of the pyramid, and the exact measurements of the base and height. 5. Studied their design and considered any math problems related to building it. 6. Solved math problems related to area. 7. Calculated how much material they would need and the cost involved. 8. Gathered materials and necessary tools. 9. Laid down the first layer of stones to make a foundation. 148 MATH 7 etext Open School BC Module 3, Section 2

151 10. Checked that the foundation was perfectly level. The foundation needed to support the whole structure, so it must have been designed to be perfectly level, and able to support the enormous weight without sinking. Lesson 3.2D Thinking Space 11. Continued to build until the pyramid was complete. The Great Pyramid we see today is the product of their well-thought out plan! It is interesting that the Egyptians chose the triangle for their design. Maybe they liked the strong qualities of a triangle. Open School BC MATH 7 etext Module 3, Section 2 149

152 Explore In Module 1, you learned how to solve problems involving the addition of integers by working your way through a series of steps: Thinking Space Step 1: Step 2: Step 3: Step 4: Understand the problem: What information am I given? Make a plan to solve the problem: How can I use the information to solve the problem? Carry out the plan to solve the problem. Answer the question asked. Instead of using these steps to add integers, how can you use them to solve problems with area? Example 1: Tim has cleared some farmland in the shape of a parallelogram. He would like to plant flax seed on his field. The seed costs $0.50 per square metre. Tim is willing to spend $100 but is worried that the seed might be too expensive. He measured his field and determined that the base is 21 m and a height is 9 m. Can Tim buy the flax seed with $100 or is it too expensive? Step 1: Understand the problem. Highlight the clue words: Tim has cleared some farmland in the shape of a parallelogram. He would like to plant flax seed on his field but the seed costs $0.50 per square metre. Tim is willing to spend $100 but is worried that the seed might be too expensive. He measured his field, and determined that the base is 21 m and a height is 9 m. Can Tim buy the flax seed with $100, or will it cost more than $100 to plant the area of farmland? 150 MATH 7 etext Open School BC Module 3, Section 2

153 Draw a picture: Lesson 3.2D Thinking Space 9 m 9 m 21 m Underline the questions: Can Tim buy the flax seed with $100, or will it cost more than $100 to plant the area of farmland? Step 2: Make a plan. Consider units: In this case, both base and height are in metres. Determine which side will be the base: In this case, base and height are given. Decide on how many parts you need to solve in the problem: This is a three-part question: 1. I need to find the area of the field. 2. I need to find the cost of planting the flax seed. 3. I need to decide if it is more or less than $100. Open School BC MATH 7 etext Module 3, Section 2 151

154 Step 3: Carry out the plan. Part 1: Write the equation. Area of a parallelogram = base height (bh). Thinking Space Area = 21 9 Area = 189 m² Part 2: The seed is $0.50 per square metre, so the cost will be the area of the field multiplied by $0.50. Cost = = $94.50 Part 3: The cost of $94.50 is less than $100, so Tim will be able to plant his field. Step 4: Answer the questions asked. Can Tim buy the flax seed with $100, or will it cost more than $100 to plant the area of farmland? Answer: Yes, Tim can purchase the flax seed and plant his field. 152 MATH 7 etext Open School BC Module 3, Section 2

155 Example 2: Robyn has dug out a garden in the shape of a triangle. She wants to plant as many daffodils as she can. The base of the garden is 240 cm and the height is 3.5 m. She knows that she can plant 35 daffodils per square metre of garden. How many daffodils can she plant? Lesson 3.2D Thinking Space Step 1: Understand the problem. Highlight the clue words: Robyn has dug out a garden in the shape of a triangle. She wants to plant as many daffodils as she can. The base of the garden is 240 cm and the height is 3.5 m. She knows that she can plant 35 daffodils per square metre of garden. How many daffodils can Robyn plant in her garden? Draw a picture: 3.5 m 240 cm Underline the questions: How many daffodils can she plant in that area of garden? Open School BC MATH 7 etext Module 3, Section 2 153

156 Step 2: Make a plan. Consider units: Base = 240 cm Thinking Space Height = 3.5 m Convert base to metres: 240 divided by 100 = 2.4 m, or 3.5 m multiplied by 100 = 350 cm Let s use b = 2.4 m and h = 3.5 m. Determine which side will be the base: Base and height are given. Decide on how many parts you need to solve in the problem: Okay, this is a two-part question: 1. I need to find the area of the garden. 2. I need to find the number of daffodils I can plant in that area. Step 3: Carry out the plan. Write the equation: Part 1: Area of a triangle = base height 2 Area = ( ) 2 Area = Area = 4.2 m² 154 MATH 7 etext Open School BC Module 3, Section 2

157 Part 2: 38 daffodils can be planted per square metre. # of daffodils = # of daffodils = 147 daffodils Lesson 3.2D Thinking Space Step 4: Answer the questions asked. How many daffodils can Robyn plant in her garden? Answer: Robyn can plant 147 daffodils in her garden. Open School BC MATH 7 etext Module 3, Section 2 155

158 Practice 1. Nick makes a rectangular sign that measures 30 cm 60 cm. He wants to attach a border that measures 50 cm 80 cm. What is the area of the border around the poster? 30 cm 50 cm 60 cm 80 cm 2. Mike wants to make a poster, and has chosen a rectangular piece of paper to use. He accidentally knocks over a bottle of paint that spills all over the corner of the paper. He will need to cut it off before starting. The poster now looks like the figure below. What is the area of the poster? 38 cm 30 cm 50 cm 156 MATH 7 etext Open School BC Module 3, Section 2

159 3. Wendy wants to build a patio covered with mosaic tiles. The patio is in the shape of a parallelogram, as shown below. The height will be 4 m and the base 300 cm. Wendy can buy 8 mosaic tiles per square metre. The cost of each tile is $0.50. How many tiles will Wendy use to cover her patio? How much will the tiles cost in total? Lesson 3.2D 4 m 300 cm 4. Robyn has dug out a garden in the shape of a triangle to plant daffodils. The height of her garden is 3.5 m. She wants to build a pathway along the side of her triangular shaped garden as shown below. She has enough pebbles to fill in 4.2 m² for a pathway. What is the base of the pathway? 3.5 m b? 240 cm Turn to the Answer Key at the end of the Module and mark your answers. Open School BC MATH 7 etext Module 3, Section 2 157

160 158 MATH 7 etext Open School BC Module 3, Section 2

161 Section Summary Summary 3.2 In this section, you learned how to measure things. You also discovered how to find the area of a triangle and a parallelogram. Applying these new skills in finding the areas of polygons, you were able to follow steps to solve problems involving area. Units of Length 10 millimetres (mm) = 1 centimetre (cm) 10 cm = 1 decimetre (dm) 100 cm = 1 metre (m) 1000 m = 1 kilometre (km) Determining the Base and Height of Triangles Equilateral Triangle h b Isosceles Triangle h b Scalene Triangle h Right Triangle h b Acute Triangle h b Obtuse Triangle b b h Open School BC MATH 7 etext Module 3, Section 2 159

162 Finding the Area of a Triangle Area of a triangle = bh 2 Finding the Area of a Parallelogram 1. Any side of a parallelogram can be the base. 2. The height of a parallelogram is a perpendicular line drawn from the base to the opposite side. 3. The height can sometimes be drawn outside the parallelogram. Here is an example of determining the base and height of a parallelogram: h b Area of a parallelogram = b h Using Steps to Solve Problems with Area Step 1: Understand the problem. Highlight the clue words: Determine if height and base are given. Determine what information is important to help you solve the problem and what is not. Eg. Side length is not needed to solve the area of a parallelogram or triangle. Draw a picture: It always helps to draw a picture so you can see what you are trying to do. Underline the questions: Figure out exactly what you are being asked. For example, are you being asked to find the length of the base or the area of a triangle? 160 MATH 7 etext Open School BC Module 3, Section 2

163 Step 2: Make a plan. Summary 3.2 Consider units: Determine if the height and base are in the same unit. If the units are different, you will have to convert the units of one to the other. For example, if height is in centimetres and base is in metres, convert the base to centimetres by multiplying by 100, or convert the height to metres by dividing by 100. Determine which side will be the base. Decide how many parts the problem asks you to solve. Step 3: Carry out the plan. Write the equation: Area of a parallelogram = base height (bh) Area of a triangle = ½ base height (½bh) Solve equation. Step 4: Answer the questions asked. Open School BC MATH 7 etext Module 3, Section 2 161

164 162 MATH 7 etext Open School BC Module 3, Section 2

165 Section Challenge The local recycling centre has asked you to design a poster for them. Part of the challenge is to use the least amount of paper. The recycling centre has a saying, less is more! They would like the poster to be in the shape of one of the triangles or parallelograms shown below. Can you choose the shape that would use the least amount of paper? Challenge 3.2 Instructions are provided for you here, as well as space to complete your work. Feel free to use your own paper if you need more space. Note: To solve this problem, you can follow the steps in the Section 2 Summary. The local recycling centre has asked you to design a poster for them. Part of the challenge is to use the least amount of paper. The recycling centre has a saying, less is more! They would like the poster to be in the shape of one of the triangles or parallelograms shown below. Choose the shape that would use the least amount of paper. To get full marks for this question, you need to show your work for calculating the area of each shape. a. b. 14 cm 0.12 m 16 cm 15 cm c. d. 18 cm 0.14 m 13 cm 8 cm e. 0.4 m 25 cm Open School BC MATH 7 etext Module 3, Section 2 163

166 164 MATH 7 etext Open School BC Module 3, Section 2

167 Section 3.3: Circles 3 Contents at a Glance Pretest 167 Section Challenge 172 Lesson A: What s in a Circle? 173 Lesson B: What is Pi? 187 Lesson C: Circumference of a Circle 197 Lesson D: Area of a Circle 211 Section Summary 225 Learning Outcomes By the end of this section you will be better able to: identify the components and characteristics of circles. describe the relationships between the parts of a circle. construct circles using geometric tools. apply a formula to find the circumference and area of a circle. solve problems that involve finding the circumference or area of a circle. solve problems involving the area of triangles, parallelograms, and/or circles. Open School BC MATH 7 etext Module 3, Section 3 165

168 166 MATH 7 etext Open School BC Module 3, Section 3

169 Pretest 3.3 Pretest 3.3 Complete this pretest if you think that you already have a strong grasp of the topics and concepts covered in this section. Mark your answers using the key found at the end of the module. If you get all the answers correct (100%), you may decide that you can omit the lesson activities. 1. Fill in the blanks: a. The is a point inside a circle from which all points on the circumference are the same distance. b. The is the longest chord that can be drawn inside a circle. c. The diameter is times as large as the radius. d. The is the distance around the circle. e. A diameter is a line segment drawn from one point on a circle to another point that passes through the. f. Every point on the circumference is at an distance from the center of the circle. g. The divides a circle in half. Open School BC MATH 7 etext Module 3, Section 3 167

170 2. Draw a circle with a radius of 2 cm. 3. Draw a circle with a diameter of 5 cm. 4. Determine the diameter of each circle with a given radius: a. r = 3.0 cm b. r = 4.6 m 168 MATH 7 etext Open School BC Module 3, Section 3

171 5. Determine the radius of each circle with a given diameter: Pretest 3.3 a. d = 4.0 m b. d = 12.4 cm 6. Determine the circumference of a circle with each diameter. Give your answer to the nearest tenth. a. d = 10.0 cm b. d = 6.7 m Open School BC MATH 7 etext Module 3, Section 3 169

172 7. Determine the circumference of a circle with each radius. Give your answer to the nearest tenth. a. r = 5.0 cm b. r = 2.3 m 8. A goldfish swims around a circular pond 4 times. The pond has a diameter of 2.0 meters. Approximately how far does the fish swim in total? Your answer should be rounded to one decimal place. 170 MATH 7 etext Open School BC Module 3, Section 3

173 9. What is the area of each circle? Pretest 3.3 a. r = 4.0 cm b. d = 6.0 m 10. A music CD has an outside diameter of 12 cm and an inside diameter of 2 cm. What is the area of the label that would fit onto the music CD? 2 cm 12 cm 11. A car wheel has a diameter of 50 cm. If the car wheel makes 1000 rotations, how many metres will the car have travelled? Turn to the Answer Key at the end of the Module and mark your answers. Open School BC MATH 7 etext Module 3, Section 3 171

174 Section Challenge Imagine you have a circular mirror hanging on your wall. Accidentally, you knock it off the wall and it falls to the ground and cracks. The wooden rim that holds the mirror in place also breaks. Oh no! What can you do? You want to replace the mirror, so you get out your measuring tape and measure the diameter of the mirror and the diameter of the rim. Luckily, both the mirror and rim hold together well enough so that you can measure the diameter of both. When you get to the store and ask for help from the person at the service counter, they ask for the area of both the mirror and rim. The store can replace the mirror and rim if you can provide the area. The person at the counter gives you a pencil, paper, and calculator. In this section, you will learn how to find the radius, diameter, and circumference of a circle. You will also learn how to determine the area of a circle and be able to provide the area of both the mirror and rim. If you re not sure how to solve the problem now, don t worry. You ll learn all the skills you need to solve the problem in this section. Give it a try now, or wait until the end of the section it s up to you! 172 MATH 7 etext Open School BC Module 3, Section 3

175 Lesson 3.3A: What s in a Circle? Lesson 3.3A Student Inquiry Radius & diameter! This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 3 173

176 Student Inquiries How are the diameter and radius of a circle calculated? How do I draw a circle with a given radius or diameter with a compass? BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: answer example answer example 174 MATH 7 etext Open School BC Module 3, Section 3

177 Lesson 3.3A: What s in a Circle? Introduction Circles are everywhere around you. Look around and see how many objects are in the shape of a circle. Dinner plates, bicycle tires, and clocks are all made in the shape of a circle. Or how about looking at the centre of a daisy it is circular. Circles are considered the perfect shape and have great significance in math, science, nature, culture, and art. Lesson 3.3A Thinking Space In many cultures, the circle represents the cycle of life as well as peace and harmony. After all, the Earth and Sun are both circular in shape. The circle is used to show that everything is connected. The circular medicine wheel and dream-catcher have healing significance to First Nations cultures. The circular yin-yang symbol represents balance in Chinese culture. A long time ago, you couldn t use a clock to tell time; you had to use a sundial and the moving shadow of the sun. The sundial was designed and built based on the angles of a circle. In this lesson, you will learn more about circles and how to draw them. You will also learn about the angles within a circle. Good luck!? Do you have any questions about how a sundial works? Explore Online Looking for more practice or just want to play some fun games? If you have internet access, go to the Math 7 website at: Look for Lesson 3.3A: What s in a Circle? and check out some of the links! Open School BC MATH 7 etext Module 3, Section 3 175

178 Warm-up Look around you and see if you can spot circular shapes. Can you think of anything that is in the shape of a circle? Remember circles come in many forms: spheres (eg. Earth, bouncy balls), discs (eg. dinner plates), spirals (eg. snail shells), cylinders (eg. soup cans), cones (eg. ice cream cones) and domes (eg. igloos). In your thinking space, list three circles that you see. Thinking Space What are some of the things that you know about circles? For example, a circle has no corners. Using the web below, write down everything you know about circles. What I Know About Circles 176 MATH 7 etext Open School BC Module 3, Section 3

179 Can you think of any similarities and differences between a polygon and a circle? Write down one similarity and one difference in your thinking space. Drawing a picture might help. Lesson 3.3A Thinking Space CIRCLES VS. POLYGONS Similarities They are both closed plane figures. They are both geometrical shapes. Differences A circle is round and has no corners. A polygon is constructed of straight sides. Connecting points with line segments make polygons. Drawing a curve that is always equal distance from a centre point makes a circle. Polygons are labelled with beginning to end points. Circles have no beginning and no end. Open School BC MATH 7 etext Module 3, Section 3 177

180 Explore Here s some circle vocabulary you ll need for this section: Thinking Space TERM DEFINITION ILLUSTRATION Centre the given point inside a circle from which all points on the edge of the circle are the same. Radius the length of a line segment drawn from the centre point to a point on the circumference. r Radii Chord plural form of radius, or more than one radius. a line segment drawn from one point to another point on the circumference. Diameter Circumference a line segment drawn from one point on a circle to another point that passes through the centre. The diameter is the longest chord that can be drawn within a circle. It also divides a circle in half. distance around the circle. r r { d C Arc a part of the curve of the circle. Equidistant Central angle equal distances. Every point on the circle is equidistant from the centre of the circle. Therefore, all radii within a given circle are the same length. an angle formed by two radii of a circle. The vertex of the angle is at the centre point of the circle. / / / 180º 30º 60º / 90º 178 MATH 7 etext Open School BC Module 3, Section 3

181 Using a compass to draw a circle with a given radius Imagine that there is a drawing contest for an upcoming Winter Olympics. The contest involves drawing a poster that highlights the Olympic Rings. You want to make sure that each circle is the exact same size for the poster. You try and draw some circles freehand, but find that each circle is not perfectly round, and are all different sizes. You get a compass to draw the circles but are not sure how to use it to so that the radius of each circle is exactly 2 cm. Here are some steps to follow: Lesson 3.3A Thinking Space How to Draw a Circle with a Compass Step 1: Step 2: Step 3: Step 4: With your pencil and ruler, draw a line segment that is 2 cm in length. Place the tip of the compass on one endpoint of the segment and its pencil on the other endpoint. Hold the compass (tighten the compass thumbscrew) so that the distance between the point and the pencil doesn t change. Put the point at the place where the center of the circle should be, and move the compass around to draw a circle. 2 cm Using your compass, try drawing circles with radii: 2 cm, 3.5 cm, and 15 mm. Open School BC MATH 7 etext Module 3, Section 3 179

182 Using a Compass to Draw a Circle with a Given Diameter The diameter of a circle is a line segment drawn from one point on a circle to another point that passes through the centre. The diameter has three important points (edge circle edge). It starts at the edge of the circle, passes through the centre, and ends at the edge of the circle. Thinking Space d If the radius is the distance from the centre to the circumference, how much longer is the diameter compared to the radius? In your thinking space, draw a circle with a radius of 2cm. Using your ruler, draw a line segment from one point on the circumference to another point that passes through the centre. Measure the diameter. How much longer is it than the radius? You should have measured the diameter as 4 cm. The diameter is always twice the length of the radius. If you know the radius, you can multiply it by 2 to get the diameter. Diameter = Radius 2 How do you calculate the radius if you know the diameter of a circle? Look at the following circle; it has a diameter of 3.0 cm. How long is the radius? Measure the radius with your ruler to confirm your answer. 180 MATH 7 etext Open School BC Module 3, Section 3

183 You can also calculate the radius by using a formula. Radius = Diameter 2 Lesson 3.3A Thinking Space Answer: Radius = Diameter 2 Radius = 3.0 cm 2 Radius = 1.5 cm Let s draw a circle with a given diameter. Step 1: Step 2: Divide the given diameter by 2 to figure out the radius. (Example, 3.0 cm 2 = 1.5 cm) Draw a line segment that is the length of the radius. 1.5 cm Step 3: Step 4: Step 5: Step 6: Place the tip of the compass on one point of the line segment and your pencil on the other point. Hold the compass so that you won t change the distance between the compass point and the pencil when you draw the circle. Draw a circle. Measure the diameter with your ruler to confirm. 1.5 cm Open School BC MATH 7 etext Module 3, Section 3 181

184 Drawing a Circle with a Rope The Ancient Egyptians used a rope compass and straight edge to build the pyramids. A rope compass acts in the same way as the compass you find in your geometry set. The Egyptians would cut a piece of rope the same length as the radius of a given circle. A person (P1) would hold one end of the rope and stand still. The other person (P2) would rotate around the person and draw the circle (in the sand most likely). The distance from the person standing still (P1) and the person rotating around (P2) remains the same. Thinking Space P1 P2 Here are the steps to follow: (do this activity on a workbench, corkboard, or cutting board not your dining room table!) 1. Draw a line segment the length of the given radius. 2. Cut a piece of string (or rope) longer than the line segment. 3. Tie a loop at the end, so the remaining string or rope is the same length as the line segment. Put the pencil inside the loop. 4. Tie the other end of the string to a pushpin. 5. Push the pin into a piece of paper to act as the centre point. 6. Move the pencil away from the pin until the string is held tight between the pencil and pin. 7. Move your pencil around the pin to draw the circle, keeping the string tight. 182 MATH 7 etext Open School BC Module 3, Section 3

185 Practice Lesson 3.3A 1. Fill in the blanks: a. The is a point inside a circle from which all points on the circumference are the same distance. b. The length of a line segment drawn from the centre point to a point on the circumference is called the. c. Radii = Plural form of. d. A line segment drawn from one point to another point on the circumference is called a. e. The is the longest chord that can be drawn within a circle. f. The diameter is times as long as the radius. g. The is the distance around the circle. h. The diameter is a line segment drawn from one point on a circle to another point that passes through the. i. A part of the curve of the circle is called an. j. Every point on the circumference is at an distance from the centre of the circle. k. The divides a circle in half. Open School BC MATH 7 etext Module 3, Section 3 183

186 2. Draw a circle with a radius of 3 cm. 3. Draw a circle with a diameter of 4 cm. 4. Using your protractor, practise measuring the angles within each circle and fill in the following table. Circle A: Circle B: A A B B C Circle C: B A C D 184 MATH 7 etext Open School BC Module 3, Section 3

187 CIRCLE ANGLES SUM OF ANGLES Circle A A = A + B = Lesson 3.3A B = Circle B A + B + C = A = B = C = Circle C A + B + C + D = A = B = C = D = 5. Give the diameter for each circle. a. r = 11 cm b. r = 6 cm c. r = 1.2 cm Open School BC MATH 7 etext Module 3, Section 3 185

188 6. Give the radius for each circle. a. d = 5 cm b. d = 14 cm c. d = 6.2 cm Turn to the Answer Key at the end of the Module and mark your answers. 186 MATH 7 etext Open School BC Module 3, Section 3

189 Lesson 3.3B: What is Pi? Lesson 3.3B Student Inquiry π This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 3 187

190 Student Inquiries What is the relationship between the circumference and the diameter of a circle? BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: answer example What is pi? answer example 188 MATH 7 etext Open School BC Module 3, Section 3

191 Lesson 3.3B: What is Pi? Introduction Since ancient times people have noticed that there is a relationship between the distance through the middle of a circle (its diameter) and the distance around a circle (its circumference). Lesson 3.3B Thinking Space Documents from as early as 240 B.C.E. tell us that Egyptian, Chinese, and Hinda mathematicians worked on the problem of defining this relationship. In this lesson you will explore question of how diameter and circumference are related. You will be able to answer the question What is pi? Explore Online Looking for more practice or just want to play some fun games? If you have internet access, go to the Math 7 website at: Look for Lesson 3.3B: What is Pi? and check out some of the links! Open School BC MATH 7 etext Module 3, Section 3 189

192 Warm-up In the second part of this lesson s activity you will be dividing numbers and rounding your answer. Thinking Space Use your calculator to answer these division problems. Round your answer to two decimal places. For example, when you ask your calculator to do: 51 7 the answer it gives is: You only want two decimal places in your answer. Look at the third decimal place. The number there is 5. You should round up at the second decimal place. Your rounded answer is: MATH 7 etext Open School BC Module 3, Section 3

193 Lesson 3.3B Open School BC MATH 7 etext Module 3, Section 3 191

194 Explore Gather the supplies you will need for this lesson s activity. Thinking Space 3 circular objects (for example: a coffee mug, a soup can, a tealight candle) string that is long enough to go around the largest of your circular objects tape marker pencil ruler or measuring tape marked in centimetres paper Step 1: Tape two pieces of paper together along the short side. Write the name of your first object on the top of the page. Use your ruler to draw a long straight line. Mark 0 at the left end of your number line. Soup Can 0 Step 2: Use the string to measure the diameter of one of your objects. Remember: Diameter is measured edge-centreedge in a straight line. Use the marker to record the diameter of your object on the string. 192 MATH 7 etext Open School BC Module 3, Section 3

195 Step 3: Transfer your measurement of the diameter to your number line. Hold one mark up to the 0 on your number line. Put 1 on your number line at the location of the second mark. Mark 2, 3, and 4 on your number line in the same way. Attach a third piece of paper if you need a longer number line. Lesson 3.3B Thinking Space Soup Can Step 4: Wrap the string around the same object and measure the circumference. Mark the string to show the circumference. Step 5: Transfer your circumference measurement to the number line. Hold one mark up to the 0 on your number line. Put * on your number line at the location of the second mark. Get two more pieces of paper and do this activity again with your second object. Get two more pieces of paper and do this activity one more time with your last object. Arrange your three number lines so you can see all of them at once. Open School BC MATH 7 etext Module 3, Section 3 193

196 Do you notice something about the location of * on each number line? It is always in the same place! Since * is always in the same place, it has a special name. Thinking Space It is π, the Greek letter Pi. The 2 on your number line shows the length of something that is 2 diameters long. The 4 on your number line shows the length of something that is 4 diameters long. The π on your number line shows the length of something that is π diameters long. Archimedes, Tsu Ch ung-chih, Aryabhata, and other ancient mathematicians wanted to know the exact value of π; its precise position on the number line. Is it close to 3 ¼? Is it more than 3.1? A lot of people spent a lot of time trying to figure out the value of π. Some of the most well-known efforts are listed at the beginning of the next lesson. That means the circumference is exactly ϖ diameters long! Can you figure out a value for π? Get a calculator. Step 1: Look at the number line for your first object. Measure (in cm) the distance from 0 to 1 on your number line. This is the diameter of your first object. Record your measurement in the chart. Step 2: Measure (in cm) the distance from 0 to * on your number line. This is the circumference of your object. Record your measurement in the chart. Step 3: How many diameters are in one circumference? Use your calculator to divide circumference by diameter. Round your answer to two decimal places (just like you did in the warm-up). This is your estimate for π. Record your estimate in the chart. 194 MATH 7 etext Open School BC Module 3, Section 3

197 Repeat these three steps with your other two numbers. Were all of your estimates for π close to each other? Start the next lesson to find out how close you were to the actual value for π. Lesson 3.3B Thinking Space Open School BC MATH 7 etext Module 3, Section 3 195

198 196 MATH 7 etext Open School BC Module 3, Section 3

199 Lesson 3.3C: Circumference of a Circle Lesson 3.3C Student Inquiry C This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 3 197

200 Student Inquiries What is the ratio of the circumference to the diameter for any circle? Can I show that circumference divided by diameter (or C d) is approximately 3.14? How can I solve problems involving circles using circumference, diameter, and radius? BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: answer example answer example answer example 198 MATH 7 etext Open School BC Module 3, Section 3

201 Lesson 3.3C: Circumference of a Circle Introduction In the last lesson, you discovered that the ratio of the circumference to the diameter of a circle is approximately 3:1. It didn t matter if the size of the diameter of the circle increased or decreased. The ratio remained the same. Even if you measure the diameter around a bouncy ball or around the planet Earth, the circumference is always a bit more than 3 times larger than the diameter. This ratio of circumference to diameter represents a constant value called pi. It is one of the most fascinating values in mathematics. Lesson 3.3C Thinking Space In this lesson, you will learn more about the importance of pi (π) and why it has fascinated and puzzled mathematicians for over 4000 years. You will need a compass and ruler to complete the activities in this lesson. Explore Online Looking for more practice or just want to play some fun games? If you have internet access, go to the Math 7 website at: Look for Lesson 3.3C: Circumference of a Circle and check out some of the links! Open School BC MATH 7 etext Module 3, Section 3 199

202 Warm-up To get ready for this lesson, let s do a quick review of the parts of the circle. Label the following diagram by filling in the blanks. Turn to the Answer Key at the end of the Module and mark your answers. 200 MATH 7 etext Open School BC Module 3, Section 3

203 Explore A Piece of the Pi The value of π has fascinated mathematicians for over 4000 years. Many people have tried to find the exact value of π. Lesson 3.3C Thinking Space π is 3.14 TIMELINE OF PI 1900 BC A Babylonian tablet (ca BC) indicates a value of for π BC An Egyptian papyrus (1650 BC) shows evidence of calculating π as BC Astronomer Yajnavalkya from India calculates π correct to two decimal places. 250 BC Mathematician Archimedes of Syracuse, Greece, works out the value of π to 4 decimal places AD Chinese mathematician, Zu Chongzhi, calculates π to 9 decimal places ( ) AD Persian mathematician, Jamshid al-kashi, finds π to 16 decimal places AD Dutch mathematician, Leudolph van Ceulen, spends much of his life calculating π up to 35 decimal places. The numbers 2, 8, 8 are engraved on his tombstone, which are the thirty-third, thirty-fourth, and thirty-fifth decimal places of π AD Welsh mathematician, William Jones, is the first person to use the Greek letter π AD English mathematician, William Shanks, takes 15 years to calculate π to 707 digits. Due to a calculation error, he is only correct up to the first 527 places AD Hungarian mathematician, John von Neumann, uses the ENIAC computer to compute π to 2037 digits, which takes 70 hours. The ENIAC was America s first electronic computer. It was so large that it weighed 27 tonnes, and was the size of an entire room AD American mathematicians, John Wrench and Daniel Shanks, find π to 100,000 digits. Open School BC MATH 7 etext Module 3, Section 3 201

204 1989 AD The Chudnovsky brothers, born in Kiev, Ukraine, find π up to the one-billionth digit on their home built super-computer AD Japanese mathematicians, Kanada and Takahashi, calculate π to the 51.5 billion digit in just over 29 hours, at an average rate of nearly 500,000 digits per second! Thinking Space Whew! You can see that pi has had a long and fascinating history. Can you guess why many mathematicians call March 14 π day? You guessed it 3.14 (3 third month, 14 fourteenth day) Calculate the Circumference given the Diameter of a Circle The circumference of a circle is always 3.14 times larger than the diameter. This is true for any circle. If you know what the diameter of a circle is, can you figure out what the circumference will be? In your thinking space, see if you can find the circumference of a circle with a diameter of 3 cm. To find the circumference, you multiply the diameter by π. Can you write a formula for this equation? The formula is written: Example: C = πd Find the circumference of a circle with a diameter of 3.0 cm. d = 3 cm 202 MATH 7 etext Open School BC Module 3, Section 3

205 Step 1: Write down the formula, and include what you know. C = π d C= π 3.0 Lesson 3.3C Thinking Space Step 2: You can press the π button on your calculator and multiply by 3, or You can multiply = 9.42 Step 3: Write down the circumference to one decimal place (because the diameter is given to one decimal place) and include units. C = 9.4 cm Calculate the Circumference of a Circle given the Radius The radius is the length of a line segment from the centre of the circle to the circumference of a circle. All radii within a circle are equal. / / / / If the diameter of a circle is 4 cm, what is the radius equal to? Remember, the radius is always half the diameter. Radius = diameter 2 Can you write a formula to find the circumference of a circle given the radius instead of the diameter? Remember: C = π d If diameter = radius (r) 2, then the formula must be: Circumference = 2 radius π Open School BC MATH 7 etext Module 3, Section 3 203

206 The formula is written as: C = 2πr Thinking Space Example: Find the circumference of a circle with a radius of 4.0 cm. Step 1: Write down the formula and include what you know. C = 2 π r C = 2 π 4 Step 2: You can press the π button on your calculator and multiply by 2 and then multiply by 4, or You can multiply = 25.1 Step 3: Write down the circumference to one decimal place (because the diameter is given to one decimal place) and include units. C = 25.1 cm The work on your page should look like this: C = 2πr C = 2 π 4 C = 25.1 cm Calculate the Diameter given the Circumference of a Circle The ratio of circumference to diameter is 3.14:1. You can find the circumference of any circle by multiplying the diameter by If you know what the circumference of a circle is, can you figure out what the diameter will be? In your thinking space, see if you can find the diameter of a circle with a circumference of 9.0 cm. To find the diameter, you divide the circumference by π. Can you write a formula for this equation? 204 MATH 7 etext Open School BC Module 3, Section 3

207 The formula is written: Lesson 3.3C d = C π Thinking Space Example: Find the diameter of a circle with a circumference of 9.0 cm. Step 1: Write down the formula and include what you know. d = C π 9.0 d = π Step 2: You can punch 9.0 into your calculator, then push the divide button followed by the π button on your calculator, or You can divide 9 by 3.14 = Step 3: Write down the diameter to one decimal place (because the circumference is given to one decimal place) and include units. d = 2.9 cm The work on your page should look like this: d = C π 9.0 d = π d = 2.9 cm Open School BC MATH 7 etext Module 3, Section 3 205

208 Practice 1. Measure the diameter of the following circles, and then calculate the circumference. a. d = 3 cm b. d = 2.7 cm 2. Determine the diameter of each circle with a given radius: a. r = 3.0 cm b. r = 4.6 m c. r = 1.9 mm 206 MATH 7 etext Open School BC Module 3, Section 3

209 3. Determine the radius of each circle with a given diameter: Lesson 3.3C a. d = 4.0 m b. d = 12.4 cm c. d = 9.2 mm 4. Determine the circumference of a circle with each diameter: a. d = 10.0 cm b. d = 6.7 m 5. Determine the circumference of a circle with each radius: a. r = 5.0 cm b. r = 2.3 m c. r = 7.0 mm Open School BC MATH 7 etext Module 3, Section 3 207

210 6. What is the radius of a circle with a circumference of 31.4 cm? 7. What is the diameter of a circle with a circumference of 24.0 cm? 8. Fill in the following table: RADIUS DIAMETER CIRCUMFERENCE 5.0 cm 28.3 cm 8.0 m 4.6 mm 1.6 cm 10.0 cm 7.2 m 208 MATH 7 etext Open School BC Module 3, Section 3

211 9. The radius of a pizza is 12 cm. Determine the circumference of the pizza. Lesson 3.3C Turn to the Answer Key at the end of the Module and mark your answers. Open School BC MATH 7 etext Module 3, Section 3 209

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213 Lesson 3.3D: Area of a Circle Lesson 3.3D Student Inquiry πr² This activity will help you get ready for, learn, and review the information in the upcoming lesson. When you turn this page over, you will find a chart containing the inquiry outcomes for this lesson. You may be able to answer some of these questions already! Start by writing down your thoughts before the lesson. When you finish the lesson, answer each question and give an example. Open School BC MATH 7 etext Module 3, Section 3 211

214 Student Inquiries BEFORE THE LESSON AFTER THE LESSON What I already know about this What I thought at the end: My final question: answer, and examples: Can I solve problems involving circles? answer example How do I estimate the area of a circle without using a formula? answer example What formula do I use to find the area of a circle? answer example How can I solve a problem involving the area of triangles, parallelograms, and/or circles? answer example 212 MATH 7 etext Open School BC Module 3, Section 3

215 Lesson 3.3D: Area of a Circle Introduction If you take a string and make a square shape, the area inside the square will be smaller than if you made a circle with the same string. This is why igloos are made in the shape of a dome. It takes the least amount of snow to get the most amount of room, so heat will not be wasted. Lesson 3.3D Thinking Space In this lesson, you will learn how to find the area of a circle by dividing it into triangles. Think of a round pizza that can be cut into triangular pieces of pizza. You know how to find the area of a triangle, so you will be able to estimate the area of a circle! You will need graph paper, ruler, pencil, calculator, and compass to complete the activities in this lesson. Open School BC MATH 7 etext Module 3, Section 3 213

216 Warm-up Area is the amount of surface within a shape. You know that the area inside a triangle is base height 2. If a triangle has a base of 3 cm and a height of 10 cm, you know that the area is , which equals 15 cm 2. Thinking Space You also know that the area inside a parallelogram is base height. If the base of a parallelogram is 6.2 cm and the height is 4.9 cm, then the area is 6.2 cm 4.9 cm. The area is 30.4 cm 2. A circle is round and doesn t have a base or height like the triangle or parallelogram. How can you determine the area of a circle without these measurements? Let s find out how to determine the area of a circle. 214 MATH 7 etext Open School BC Module 3, Section 3

217 Explore Estimating the Area of a Circle In the space below, brainstorm some possible ways to estimate the area of a circle. Lesson 3.3D Thinking Space Try these ways to estimate the area of a circle: Method 1. Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: You can draw a circle on graph paper and count the squares. Draw a line segment 3 units long on graph paper. This will be the radius of your circle. Place your compass point on the beginning point of the line segment and your pencil on the end point of the line segment. Draw a circle with your compass. Count the whole squares inside the circle. Combine parts of squares to equal whole numbers and add that number to your whole square count. The total number of squares inside the circle is the area. 3 units Did you count approximately 28 square units inside the circle? Open School BC MATH 7 etext Module 3, Section 3 215

218 Method 2. Estimate the area by dividing a circle into triangles. A circle is round, so it does not have base and height measurements like the triangle and parallelogram, but there is something you can do to create a base and height. Think of a circle like a pizza pie. If you cut the pizza into 8 slices, you will have 8 triangular pieces of pizza. You can then arrange the pizza slices on a table into a parallelogram. You can do the same thing with a circle. If you divide a circle into triangles and form a parallelogram, you can measure a base and height. By multiplying the base and height, you can estimate the area of a circle. Does the area of the circle change when you turn it into a parallelogram? Thinking Space Let s figure out how to divide a circle into triangles to estimate the area. Follow the steps below on a separate piece of paper. Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7: Step 8: Step 9: Draw a line segment on graph paper to be the length of the radius. For example, use a radius of 3 cm. Adjust your compass so that it is the width of the line segment. Draw a circle using the line segment as the radius. Use your scissors to cut out the circle. Fold your circle in half, then again in half and again in half. Unfold the circle to find 8 triangular segments. Cut along the fold lines to get 8 triangles. Arrange the triangles on graph paper so that it makes a parallelogram shape. Use your ruler and pencil to outline the base and height of the parallelogram. Step 10: Count the squares to measure the base and height. Step 11: Estimate the area by multiplying the base x height. Step 12: Write down your answer in units. 216 MATH 7 etext Open School BC Module 3, Section 3

219 Example: Lesson 3.3D Thinking Space r = Height = 3 squares Base = 9.25 squares Area = base height = 3 squares 9.25 squares = 28 square units = 28 units² Using a Formula to Find the Area of a Circle You know how to estimate the area of a circle by dividing a circle into triangles to make a parallelogram. By measuring the height and width of the parallelogram, you can determine the area. Look at the parallelogram below. How is the radius of the circle similar to the height of the parallelogram? r = Height = radius Base = Half of the Circumference of the circle = 2πr 2 = πr Did you guess that the height and the radius are equal? Yes, it is. How is the width of the parallelogram similar to the circumference? The width of the parallelogram is half the length of the circumference. The circumference of a circle is 2πr so the width is equal to ½ 2 π r, which is π r. The width is πr and the height is r. Can you determine the formula to find the area of a circle? How did you do? Open School BC MATH 7 etext Module 3, Section 3 217

220 Let s try together: Area of a parallelogram = base height = π r r (or r²) Area of a circle = πr² Thinking Space Let s practise using the formula to find the area of a circle. Example 1: A circle has a radius of 5.0 cm. Find the area of the circle. Step 1: Use the formula to find area. Area = π r² Area = ² Area = Area = 78.5 cm² Step 2: Write down the area with units. The area of the circle is 78.5 cm². Example 2: A circle has a diameter of 8.0 cm. Find the area of a circle. Step 1: Find the radius of the circle Radius = diameter 2 Radius = 8 2 = 4 Step 2: Use the formula to find area. Area = π r² Area = ² Area = Area = 50.2 cm² Step 3: Write down the area including units. The area of the circle is 50.2 cm². 218 MATH 7 etext Open School BC Module 3, Section 3

221 What if an object has multiple shapes added together? How can I find the area? Many objects or structures you see in your daily life are often a combination of shapes. Have you noticed that the shape of a hockey rink is a rectangle in the middle with two semi-circles at each end? Or how about the shape of a doghouse? It has a triangular roof and square bottom. You can also find circles within circles, like the shape of a CD. How can you find the area if an object has two shapes? What formula can you use? Lesson 3.3D Thinking Space To find the area of these shapes, you need to break up the question into parts and then solve. Here are some examples to show you how to solve a multiple-step problem. Example 3: This hockey rink consists of a rectangle and two semi-circles. The length of the rectangle is 16.0 m and the radius of the semi-circle at each end is 5.0 m. Find the area of the hockey rink. r = 5 m 10 m 16 m Open School BC MATH 7 etext Module 3, Section 3 219

222 Step 1: Understand the problem. a. Highlight the clue words: hockey rink consists of a rectangle and two semi-circles. length of the rectangle is 16.0 m radius of the semi-circle at each end is 5.0 m find the area of the hockey rink. Thinking Space Determine if measurements are given: Length of rectangle = 16 m Radius of semi-circle = 5 m b. Draw a picture: Look at the illustration on the previous page. Step 2: c. Underline the questions: You are asked to find the area of the hockey rink. The shape is a rectangle plus two semi-circles. Make a plan. a. Make a list of clue words, and how they will help you solve the problem. hockey rink consists of a rectangle and two semi-circles What is a semi-circle? A semi-circle is half of a circle. You know that two semi-circles equal one circle. Think of it like a pizza that you cut in half. Two halves of a pizza makes a whole pizza. One shape of the hockey rink is a whole circle, and the other shape is a rectangle. radius of the semi-circle at each end is 5.0 m You are given the radius of the two semi-circles so you can calculate the area. length of the rectangle is 16.0 m You are given the length of the rectangle but not the width. Is there any other information that can help you find the width? Look at the illustration of the hockey rink. You will see that the radius is equal to half the width of the hockey rink. The radius of the semicircle is ½ the distance of the width of the rectangle. So, the width of the rectangle equals 5 2 = MATH 7 etext Open School BC Module 3, Section 3

223 find the area of the hockey rink Add the area of the circle and rectangle to determine the area of the hockey rink. Lesson 3.3D Thinking Space b. Consider units: The radius of the semi-circle and the length of the rectangle are both in metres. c. Decide on how many parts you need to solve in the problem: This is a three-part question: 1. Find the area of the rectangle. 2. Find the area of the circle. 3. Add the area of the rectangle and the area of the circle to find the area of the hockey rink. Step 3: Carry out the plan. Find the area of the rectangle. Area = length width Area = Area = 160 m² The area of the rectangle equals 160 m². Find the area of the circle. Area = pir² A = ² A = A = 78.5 The area of the two semi-circles equal 78.5 m². Add the area of the rectangle and the area of the circle to find the area of the hockey rink. Area = rectangle + two semi-circles A = A = m² Step 4: Answer the question asked. What is the area of the hockey rink? Answer: The area of the hockey rink is m². Open School BC MATH 7 etext Module 3, Section 3 221

224 Practice 1. What is the area of each circle? a. r = 2.0 cm b. r = 2.3 m 2. What is the area of each circle? a. d = 8.0 m b. d = 3.8 cm 3. A pumpkin pie has a diameter of 28 cm. a. What is the radius of the pie? b. What is the area of the pumpkin pie? c. The pumpkin pie is cut into 4 pieces. What is the area of one piece of pie? 222 MATH 7 etext Open School BC Module 3, Section 3

225 4. Hayden is painting a circular mural on the side of a building. He will need to paint two coats of paint. The radius of the mural is 1.5 m. How much area in total will he need to paint? Lesson 3.3D 5. A music CD has an outside diameter of 12 cm and an inside diameter of 2 cm. What is the area of the label that would fit onto the CD? 2 cm 12 cm Turn to the Answer Key at the end of the Module and mark your answers. Open School BC MATH 7 etext Module 3, Section 3 223

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227 Section Summary Summary 3.3 Circle Terminology Review the vocabulary of circles. You can go back to the table in Lesson 3.3A, or review the glossary. How to Draw a Circle Step 1. Step 2. Step 3. Step 4. With your pencil and ruler, draw a line segment the length of the radius.. Place the tip of the compass on one point and its pencil on the other point. Hold the compass so that the distance between the point and the pencil doesn t change. Draw a circle. Radius and Diameter of a Circle r d Radius = Diameter 2 Open School BC MATH 7 etext Module 3, Section 3 225

228 Drawing a Circle with a Given Diameter Step 1. Step 2. Divide the given diameter by 2 to figure out the radius. Draw a line segment that is the length of the radius. 2 cm Step 3. Step 4. Step 5. Place the tip of the compass on one point of the line segment and its pencil on the other point. Hold the compass so that you won t change the distance between the compass point and the pencil when you draw the circle. Draw a circle. 2 cm Step 6. Measure the diameter to confirm. Formula to Calculate the Circumference of a Circle π is approximately equal to 3.14 Circumference = π diameter C = πd C = 2πr Formula to Calculate the Area of a Circle Area of a circle = πr² 226 MATH 7 etext Open School BC Module 3, Section 3

229 Section Challenge Challenge 3.3 Imagine you have a circular mirror hanging on your wall. Accidentally, you knock it off the wall and it falls to the ground and cracks. The wooden rim that holds the mirror in place also breaks. Oh no! What can you do? You want to replace the mirror, so you get out your measuring tape and measure the diameter of the mirror and the diameter of the rim. Luckily, both the mirror and rim hold together well enough so that you can measure the diameter of both. When you get to the store and ask for help from the person at the service counter, they ask for the area of both the mirror and rim. The store can replace the mirror and rim if you can provide the area. The person at the counter gives you a pencil, paper, and calculator. A mirror with a wooden rim has an outside diameter of 30 cm. The diameter of the mirror is 24 cm. What is the area of the mirror? What is the area of the rim? Your answer should be to one decimal place and include units. 30 cm 24 cm Open School BC MATH 7 etext Module 3, Section 3 227

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231 Module 3 Templates Templates 3 Open School BC MATH 7 etext Module 3, Templates 229

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241 Answer Key Table of Contents Answer Key 3 Pretest Lesson 3.1A Warm-up 242 Lesson 3.1A Practice Lesson 3.1A Practice Lesson 3.1B Practice 244 Lesson 3.1C Practice 245 Lesson 3.1D Practice 246 Lesson 3.1E Practice 248 Section Challenge Pretest Lesson 3.2A Practice 251 Lesson 3.2B Practice 253 Lesson 3.2C Practice 253 Lesson 3.2D Practice 255 Section Challenge Pretest Lesson 3.3A Practice 258 Lesson 3.3B Warm-up 260 Lesson 3.3C Warm-up 261 Lesson 3.3C Practice 261 Lesson 3.3D Practice 263 Section Challenge Open School BC MATH 7 etext Module 3, Answer Key 239

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243 Answer to Pretest Parallel: A, C Perpendicular: B, D Answer Key 3 2. a. DRAWING: Line segment = 4 cm, angle = 45 b. DRAWING: Line segment = 3 cm, angle = 120 c. DRAWING: Line segment = 2.5 cm, angle = Any three examples, such as streets and avenues, the corners of a picture frame, the corner of the floor and the wall, etc. 4. Any three examples, such as rails on a train track, rails of a ladder, the sides of a ruler, etc. 5. a. compass b. perpendicular bisector c. T d. protractor e. angle bisector 6. a. DRAWING: Perpendicular bisector is at 36.5 mm b. DRAWING: Perpendicular bisector is at 7 cm c. DRAWING: Perpendicular bisector is at 5.5 cm 7. a. DRAWING: Angle = 130 with angle bisector at 65 b. DRAWING: Angle = 70 with angle bisector at 35 Open School BC MATH 7 etext Module 3, Answer Key 241

244 3 cm 3 cm 8. 5 cm 5 cm 9. 6 cm 6 cm Answer to Lesson 3.1A Warm-up 1. meet 2. degree 3. parallel o clock 6. perpendicular 242 MATH 7 etext Open School BC Module 3, Answer Key

245 7. right Answer Key 3 8. acute 9. obtuse 10. protractor 11. zero 12. a. greater than b. right c. less than d. right e. less than f. greater than Answer to Lesson 3.1A Practice 1 1. a. 30 b. 65 c. 160 d a. A = 45 b. B = 10 c. C = 95 d. D = 77 e. E = 150 Open School BC MATH 7 etext Module 3, Answer Key 243

246 Answer to Lesson 3.1A Practice 2 1. Parallel: A, C Perpendicular: B, D 2. Right angles: A, C 3. a. b. 30º 45º c. d. 60º 90º e. 180º 4. Any three examples, such as streets and avenues, the corners of a picture frame, the corner of the floor and the wall, etc. 5. Any three examples, such as rails on a train track, rails of a ladder, the sides of a ruler, etc. Answer to Lesson 3.1B Practice 1. a. DRAWING: Flower w/ petals, drawn using compass b. DRAWING: 30, 45, and 60 degree angles, drawn using only triangles c. DRAWING: Star 244 MATH 7 etext Open School BC Module 3, Answer Key

247 2. Answers will vary. Answer Key 3 3. Answers will vary. 4. a. compass b. perpendicular bisector c. T d. protractor e. triangle f. ruler or straight edge g. intersection point h. angle bisector Answer to Lesson 3.1C Practice 1. Perpendicular lines meet at a 90 angle A perpendicular bisector is a line that intersects a line segment at 90 degrees and divides it into two equal lengths. 4. a. DRAWING: Perpendicular bisector is at 3.5 cm b. DRAWING: Perpendicular bisector is at 5 cm c. DRAWING: Perpendicular bisector is at 8.5 cm 5. Either or both of these answers is correct. 1. Draw a point on the perpendicular bisector. This point should be the same distance from each end of the original line segment. 2. Measure the line segment with your ruler. The perpendicular bisector should divide the line in half. Use your protractor to measure the angle between the perpendicular bisector and line segment. It should equal 90. Open School BC MATH 7 etext Module 3, Answer Key 245

248 6. BEWARE of Bears 7. Meet at the Zebra exhibit. Gate B Gate A Answer to Lesson 3.1D Practice 1. a. DRAWING: Angle bisector is at 45 degrees b. DRAWING: Angle bisector is at 15 degrees c. DRAWING: Angle bisector is at 30 degrees d. DRAWING: Angle bisector is at 25 degrees e. DRAWING: Angle bisector is at 60 degrees 246 MATH 7 etext Open School BC Module 3, Answer Key